In France, the law requires a contract for a postdoctoral researcher be established at latest 3 years after obtaining the doctoral degree, for a minimum duration of one year and a maximum of 3 years, and renewable once within the limit of a total duration of 4 years.
A contract for a postdoctoral researcher is a special contract that notably derogates from the general expectation under French law that employment should be on a permanent basis. It also provides additional requirements on the part of employer on the support and training of the postdoctoral researcher. It is subject to special rules from the government, and a postdoc employed by a public institution may have different rights than other employees of the state.
The law does not prevent you from doing what postdocs do (like she mentioned, as a non-permanent researcher), but your contract cannot be administered under special rules for postdocs, which may be a negative factor for your employer.
ERC grants in general do not prohibit hiring non-permanent scientific staff, but depending on the institution, your non-postdoc status may result in additional accounting or other bureucratic work.
In general, such restrictions are not uncommon in Europe. The rule on who qualifies as a postdoc, with associated rights (or lack thereof), can come from many sources. For example, a postdoc grant or fellowship may impose the restriction to support young researchers. Postdocs also often benefit from exemption from normal immigration rules (quotas, proof of inability to find domestic workers etc.), so a country may impose requirements to prevent abuse. For example, in Switzerland, a postdoc usually has to start within two years of graduation and usually cannot be more than four years, as a postdoc work permit is not subject to normal immigration procedure and should not be used to circumvent the law.
Similarly, the working conditions of many postdocs (like graduate assistants) are in violation of the labour laws in effect, or would be if not for their special status. Labour unions also have an interest in protecting interests of workers and (if you are cynical) receiving dues from more members. Collective agreements often set minimum salary depending on the experience of workers, and may require the classification of certain employees not as a postdoc, which has a consequent effect on the compensation and protection afforded to them.
The goal of these rules are to prevent the abuse of permanent non-permanent positions widespread in academia, although it may have an unfortunate effect for cases like yours. You may also point out the conflicting stance of the government who does not fund well enough to support more permanent positions, but that would be a separate discussion.
