Within the EU countries only accept other countries' driving license if the driving test was made in one of the EU countries. In case of exchanges the country where the test was taken will be added visibly to the license. So even if you could exchange your current one to a Hungarian one (which is only possible if it was done in a country which has an agreement for exchanges with Hungary), it will say which country the test was taken in, and the French authorities will not exchange it afterwards, as it is not from a country they allow it (unless of course they also allow licenses from that country).
You can use your license for one year with an IDP (international driver's permit) after which you need to take the exam in France.
Note that Hungary does potentially allow exchanges from all countries, but you would need to present either a residence card (lakcímkártya) or a residence permit. France has a quite long list of countries where they allow exchanges from, however they also explicitly state that exchanges are only valid if the originating country (e.g. where you took the test) is from the same list, even if the license was issued in another country, like Hungary, from an exchange.
Also note that as a student you might get away with going home every year and getting a new IDT, as sometimes you are also considered a resident in your home country, meaning your license is valid and you can get a fresh IDT. This might not be strictly legal, but definitely something that is hard to enforce, especially if you didn't have any driving infractions in the first year.