Not wishing to get drawn into a big debate over this, I will try to answer briefly.
Those who look for a specific law that says that holiday home owners or visitors are not allowed will not find one.
The control is due to the use of the word “domicile”. I know that there are quite a few dictionary definitions but when the french authorities use it “domicile” refers to your principal residence of which most of us have only one.
If you live in Croydon but own a property in Normandy for holidays & weekend breaks, your principal residence is in Croydon.
So, when you look at the government guides in their original language you will see that one of the documents you need to show is proof of domicile (not “adresse” or “residence”) & goes on to say that an EDF bill or similar will do. Of course, second home owners will have one of these but to use it as proof of domicile would be untrue when it is only proof of the address of your second home. At the prefecture it was expected that the applicant would be honest & as 99.9% of those registering a car DID live full-time in France there would be no need to question the applicant - even if the person on the desk was aware of the difference. The result is that the registration would go through.
Technology has moved & prefectures no longer directly handle applications as it is all dealt with online.
In order to access the Agence Nationale des Titres Sécurisés (National Agency for Secure Documents, as in not handed out like confetti) you need to live here which means that you declare income, you are in the french healthcare system or are registered for social security. These organisations put you on the INSEE database, similar to the UK Electorial Roll, so most second home owners will not have these so direct access would be denied making registration of a car impossible.
This also applies to new arrivals though, so it this instance it is possible to use a third party to access ANTS. The third party has to make a declaration to ANTS though, & some either do not understand or possibly care what consequences they might face in the future. This is the declaration" Rimmer Mark certifie sur l’honneur s’être assuré(e) que FARAGE NIGEL est bien domicilié(e) à la nouvelle adresse renseignée et s’engage à mettre à disposition, sur demande des services du ministère de l’intérieur, toute pièce le justifiant." & the rule which should prevent honest people doing something they shouldn’t - “J’ai bien pris connaissance que “les fausses déclarations et l’usage de fausses déclarations sont punis de 3 ans d’emprisonnement et de 45 000 euros d’amende”, article 441-1 du code pénal. Je certifie notamment que les informations renseignées relatives au régime de propriété du véhicule sont exactes.”
The requirements are therefore clear & being from the UK or owning the property does not bestow any special treatment - a french person cannot register his car to his second home (which may well be in a more expensive tax zone than his principal residence, in case someone wants to bring up the feeble “registration tax” saving theory), an Italian can’t (europa.eu Q & A " I am Italian and every year I take a 3-month holiday in my second home in the French Alps, travelling by car. Can I register my car in France?
NO — You can only register your car in the country where you are permanently resident.) so why would the Brits be exempt?
On the point of the all important use of the word “domicile” This made interesting reading
" Amended by Order n°2020-1733 of 16 December 2020 - art. 4
The election of domicile is granted for a limited period. It is renewable by right and may only be terminated under the conditions set out in Article L. 264-5.
The communal or inter-communal social action centres and the approved organisations will provide the persons concerned with a certificate of election of domicile stating the date of expiry.
The certificate of election of domicile may not be issued to a person who is not a national of a Member State of the European Union, another State party to the Agreement on the European Economic Area or the Swiss Confederation, who is not in possession of one of the residence permits provided for in Title II of Book IV of the Code on the Entry and Residence of Foreigners and the Right of Asylum, unless he or she is applying for the State medical aid mentioned in Article L. 251-1 of the present code, legal aid pursuant to the third or fourth paragraphs of Article 3 of Law No. 91-647 of 10 July 1991 relating to legal aid or the exercise of civil rights recognised by law.
In accordance with Article 20 of Order No. 2020-1733 of 16 December 2020, these provisions come into force on 1 May 2021"