Both Canada and the US still have weird quasi-states on their territory for citizens of Native American/First Nation origins. However in the 21st century this appears to be quite an archaic concept, since in theory there shouldn't be any more areas restricted to members of a certain race or ethnicity.
So why do these Indian reservations still exist in North America? Are there any plans to reform them into regular villages/towns/cities without a special status?
To remove such rights would be unfair. But while the reservations have a particular context, self-determination for ethnic minorities is not so unusual.
The Native American and First Nation peoples were moved to reservations as a deliberate policy of "ethnic cleansing" in the 19th century.
The various tribes were given limited rights to self-governance on their new reservations. Now if we propose to remove those rights, justice would expect that the lands that were taken should be returned. However this is not a practical idea: the original inhabitants are long dead, the land has been dramatically changed and there are several hundred million new people living on those lands. The Indian nations currently living on reservations want to preserve such rights as they have. There is, of course, no legal barrier to leaving a reservation, and many people of Native American and First Nation descent don't live on reservations.
As such, removing such limited rights that the Native American nations have to self-determination without giving something in return would be an extension of a historical injustice.
By way of analogy: A school bully steals a fountain pen worth $20 from another boy. Laughing at the boy the bully gives the boy $1 and says "It's not stealing, I'm buying the pen". 20 years later, and both the bully and the boy have grown up. The bully says "It was wrong of me to steal your pen and give you a dollar in exchange. So I'm going to take back the dollar."
The idea of certain peoples having limited self-determination is not so unusual. The Welsh (who were perhaps victims of an earlier ethnic cleaning by the invading Saxons) have an assembly and the Scots have a parliament. Other regions in Europe have similar arrangements. It is quite common for an ethnic minority that is geographically limited to be granted some form of limited self-determination.
James K's answer is excellent, but I want to expand further on how tribes are organized presently.
However in the 21st century this appears to be quite an archaic concept, since in theory all citizens are supposed to be equal rather than receiving certain privileges simply because of their race or ethnicity.
I know you are from Europe, so you may be unfamiliar with the history of reservations/reserves in the US and Canada, but this is a misconception of the arrangement. ("Reserves" is the Canadian term, but I'll use "reservations" because I'm American.)
It's also a little unclear which privileges you refer to. The only one I can think of associated with reservations is the ability to run their own court system, which is of course, available to every municipality, county, and state in the US, so far from limited to tribal governments. In fact, their court systems are more limited than others, as they cannot prosecute non-members.
The point is, though, that the arrangement exists not between the federal government and US citizens of certain descent, but between the US federal and state governments and the "domestic dependent nations" that represent Native American/First Nations peoples. This concept is known as tribal sovereignty.
Tribal sovereignty in the United States is the concept of the inherent authority of indigenous tribes to govern themselves within the borders of the United States.
America is sometimes considered to be a nation organized around shared ideals, rather than ethnic history, like most other countries are. (There was a good Politics question about that, but I can't find it.) However, tribal nations are, an issue that causes no end of contention within the nations (look up "disenrollment.") Even though the US is set up this way, it can still have relationships with ethnic nations, and it does.
So it's best not to consider the relationship not one between America/Canada and its citizens of Native American descent, but one between the respective governments and their dependent (ethnic) nations.
Simply put, disorganizing a reservation would be disorganizing a nation with (limited) sovereignty, and perpetrated by the nation(s) that stole their land and forced them into the reservation in the first place.
* All that said, this is very heterogeneous. British Columbia has a lot of unceded land and unsigned treaties. Alaska has a different set-up than the lower 48, which I'm not qualified to discuss, but the notion holds at a high level for why the reservations continue to exist.
However in the 21st century this appears to be quite an archaic concept, since in theory all citizens are supposed to be equal rather than receiving certain privileges simply because of their race or ethnicity.
All citizens of a nation, perhaps - and this is by no means a universally held or implemented belief. (Just ask Saudi women, people under 18, Arab Israelis, etc.)
The fundamental status of federally-recognized Native American tribes is that of a separate nation ("domestic dependent nations", i.e. with exceptions; see below), and hence, we US citizens don't get to demand the rights of another nation, just like I can't demand to vote in Mexico.
Per the US Bureau of Indian Affairs:
The relationship between federally recognized tribes and the United States is one between sovereigns, i.e., between a government and a government. This “government-to-government” principle, which is grounded in the United States Constitution, has helped to shape the long history of relations between the federal government and these tribal nations.
Tribes possess all powers of self-government except those relinquished under treaty with the United States, those that Congress has expressly extinguished, and those that federal courts have ruled are subject to existing federal law or are inconsistent with overriding national policies. Tribes, therefore, possess the right to form their own governments; to make and enforce laws, both civil and criminal; to tax; to establish and determine membership (i.e., tribal citizenship); to license and regulate activities within their jurisdiction; to zone; and to exclude persons from tribal lands.
Limitations on inherent tribal powers of self-government are few, but do include the same limitations applicable to states, e.g., neither tribes nor states have the power to make war, engage in foreign relations, or print and issue currency.
They continue to exist for a variety of reasons, including but not limited to:
In some parts, such as in British Columbia, the First Nations never signed any treaties with the Government of Canada. For example, see this page on the Kitselas Administration. Even if the government declares a particular reservation, some native tribes may decline to acknowledge that the Government of Canada has the sovereign right to do so. In British Columbia, there are regular conflicts when it comes to constructing pipelines, with First Nations tribes arguing that the Government of Canada has no right to build a pipeline on their land without their consent, partly because they never signed a treaty with the Government of Canada to begin with.
In the context of such conflicts, a Government decision to take away the little breadcrumbs that are native rights acknowledged by the Government, would almost certainly escalate the conflict. It would also violate the U.N. Declaration of the Rights of Indigenous Peoples (of which Canada was one of only four nations to oppose, although since then different governments have expressed different views on whether to adopt those rights into Canadian law).
The relationship between the Native American nations (there are many, not just one) and the US are (theoretically) at least vaguely similar to those between the nations of Europe and the European Union. Each of them is a separate nation that has signed treaties to cede some specific rights to the "larger" government entity.
There are certainly differences--the rights the Native American tribes ceded to the US federal government were largely those of things like foreign relations, while those ceded in the case of the EU treaties tended to relate more to economics and trade.
Regardless of those differences, your question is roughly similar to asking why the EU hasn't just become the sole government of the entire European continent. That answer is fairly simple: because EU only have the rights ceded to them by treaty, and that doesn't include the right to dissolve the separate nations and take over all governance of those people.
Many of the same issues arise in both cases
Nonetheless, I think in both cases, the overarching government entity currently makes at least some minimal attempt at following (its interpretation of) the treaties it signed. Unilaterally dissolving a nation it has recognized in a treaty would be a clear violation that's unlikely to happen.
There are exceptions to those rules as well, of course--in particular, there are cases where the natives did not sign any treaty ceding any rights to the larger government. I suppose that leaves a little more room for the overarching government to simply treat that nation as not existing--but it would certainly violate current expectations about how native nations should be treated to the point that most would find it unacceptable anyway.
In Europe, there are some small nations. Denmark is a postage stamp. Luxembourg is a rounding error. Why doesn't Germany just annex them?
For the same reason Canada doesn't dissolve the reserves. Historically, the multitude of native tribes are distinct nations with distinct histories. Canada has no more right to consume the reserves than Russia has to absorb Crimea.
Functionally, we can consider the reserves to be enclaves with limited sovereignty.
