No, while active duty your home of record will never change unless you formally have it changed. You will continue to pay taxes and always be a resident of your home of record state even if you own property, register vehicles, etc., elsewhere.
Different story when leaving active duty. For you no big deal since you are OK with maintaining a high-tax state as home of record. For those of us who have very low-tax home of records, when we get out we will need to prove why we should continue to pay taxes in that state when we have properties and cars registered elsewhere. Long story short, once we are off active duty we then become residents of the state in which we occupy most often according to federal law.
Home of record and residency (more correctly referred to as domicile, in a legal sense) are not interchangeable.
Home of record is a military-specific term, used to indicate your domicile at the time you entered active duty. It is typically relevant only for determing how much the military will pay to move you when you separate or retire. For officers, it is virtually impossible to change your home of record, unless you have a break in service.
Domicile is a legal term that indicates where you primarily live. For active duty personnel, this is a hypothetical concern, because it's actually just an indicator of where you intend to reside after your active duty service. It is extraordinarily difficult to gauge intent legally, which is why secondary indicators are used, to wit, where you pay state taxes, where you own property, from where you have a driver's license, where your cars are registered, where your last will and testament is to be executed, or where you have a medical license. The single biggest factor is probably where you vote.
But it's a continuum. So, to answer the OP's question - no, registering your cars in one state will not sway anyone if the preponderance of evidence otherwise points to you being domiciled in a second state.
There is not a federal law that dictates the specifics of domicile. There is just federal law providing relief to military personnel from the particulars of each state (originally the Soldiers' and Sailors' Civial Relief Act and its subsequent iterations) based upon their presence in the state for the sole purpose of military service. Servicemembers are free to change their domicile as they see fit, provided that they have the documentation to back it up. All it takes is a call/visit to the finance office.
That said, all of this is almost entirely irrelevant. As far as I can discern, the only reason why it would become relevant is if state A comes after you for unpaid taxes while you claimed to be domiciled in state B. That's a losing legal strategy, and not worth any state revenue agency's time.
