It is not normal in the sense that it is not legal. But it is normal in the sense that it does happen.
Legally speaking, the police only has the right to require identification when there has been a breakdown of public order (“trouble à l'ordre public”), or in the course of a judicial inquiry. For example, ID checks in the vicinity of a riot are legal, but not in the vicinity of a peaceful demonstration. In addition, the government.
In practice, though, civil rights associations regularly have an uphill battle contesting the validity of identity checks. A decision by a court five years later that the police acted unlawfully won't help you much.
The law in France requires that everyone be able to justify their identity. Furthermore, people who are not citizens must be able to justify that they are in France lawfully. So you should always carry your passport, EU identity card, carte de séjour, or whatever document shows that you have the right to be in the country. There is no fine for a violation, but if you can't justify your identity, the police may bring you to a police station and hold you for several hours.
Sadly, but unsurprisingly, if you're Black or Arab/North African, the police is more likely to give you a hard time. Note to North Americans: in French (and more generally European) racist categorization, “Hispanic” is not a thing, but “Arab” is.
The border police may perform identity checks near borders. This includes a 20-km zone near borders with other Schengen countries, even when systematic checks are not in place. This also includes airports and international trains and even train stations where international trains call.
If you're driving, then you must have your driving license and the vehicle's papers (including proof of insurance). Unlike pedestrians, the police may legally stop any vehicle at any time and request the driver's and the vehicle's papers.