It is not obligatory to register a foreign vehicle in France unless the owner is resident in France. A resident is someone who is domiciled in France for more than six months (183 days) per year, who is employed in France, or has made a recognised and deemed permanent move toward residency in France (ie has no domicile in the country of origin, severed other ties with that country that indicate an intention to leave it with permanence). Under EU law, a private vehicle may be temporarily imported and used on French roads for up to six months in any 12 month period. The vehicle must be re-registered in France if it is owned and used by a resident of France. It is against the law for the resident of an EU country, while in that country, to drive any vehicle registered in another EU country. So a person who is a resident of France may only drive a French-registered car while they are in France (with the exception of cross-border workers). An EU-registered vehicle must satisfy the legal roadworthiness requirements of its country of registration to be legal to drive elsewhere in the EU.
There will be some canteen lawyers who will try to add or detract from this I have no doubt, I can only say that I have practiced law for over thirty years and the above is my understanding of the agreed EU regulations.
There seems to be some misunderstanding as to what qualifies a person to be classified as a French resident - the legal standpoint, in my experience, is that even if you have only been in France for one week, if your intention is to be a permanent resident and you have made efforts to that end (selling your UK home, terminating tax responsibilities, etc) then you are deemed to be a resident of France per se. From that point you must register your UK vehicle in France. It is also to be noted that you will nullify your UK car insurance and MOT the moment you enter France with the intention of being a permanent resident (unless you have made provision with your insurers beforehand).
The 183 day (6 month) limit has been set to deal with people who spend lengthy periods in more than one country and abuse different legislation to avoid various local taxes. The UK road tax is a perfect example, a French registered vehicle owned by a UK resident pays no road tax in the UK, that person may only actually spend a very small period of time in France, therefore the Law states if that person spends more than 183 days in the UK then they are deemed to be a UK resident and the vehicle should be on UK plates and paying UK road tax, not hiding that responsibility in France - and that 183 days need not be continuous but in total.
The same applies for UK persons now resident in France who fail to get a CT, French insurance and a Carte Grise for their UK vehicles - should they be involved in a serious accident, or God forbid a fatal accident, and the authorities can prove residency then be assured lengthy prison sentences will ultimately follow (and residency can be proved by obtaining details of services paid for, EDF, France telecom, bank account details, Tax Foncier/Habitat etc.)