Pretty much what the title says, I was wondering because although I 'm using Graphene OS, I have to use Android Auto for many reasons (I 'm not gonna bother you with a list of reasons). For the time being I m using an old phone always on my car just for the AA, but this is a temporary solution and day by day this phone maybe die or slow down to a crawl.
My answer would be not very private. On stock, Google controls all the system apps, and has Google Play Services spying on you as well. You can probably reduce all of this, by denying network access to all of them (assuming you won’t use Google Play Services). But, using the OS of a company known to spy on people, is not going to be private, regardless of whether or not you have an account.
I think it is difficult to say with any certainty considering that a lot of the stock pixel apps are proprietary (opaque) and developed by a data harvesting and advertising company, and that some of those apps are privileged apps and hooked into many different aspects of the system.
I think that whether you use a Google account or not will have not have a big effect on privacy (privacy and anonymity are different concepts, and not using an account is more related to the latter). But it could still have positive impact on what data Google can tie to your true identity. But it is likely that if Google cares enough about establishing that link, it wouldnt be difficult, even without a Google account, unless you are extremely disciplined and cautious (but I want to be clear this is just speculation)
It is a bummer that there is no open-source Android Auto alternative, and that Android Auto doesnt work on GOS, it seems this is one of the few remaining barriers to using/recommending GOS without reservation that needs to be solved.