I know I already went but seems digging deeper is in order.
Plenty of companies made themes that did not have games, DLC or anything like that available. Would have been similar idea to having posters or whatever for the kids in the eyes of marketing slimes (better yet from Microsoft's new piece of tech that surely can't fail).
Wikipedia has long been dubious, maybe not quite as incomplete as the likes of gamefaqs used to be for Japanese games, different names for the same game between regions being unconnected and release dates for non NA releases, and things like metacritic and opencritic still are for a lot of that (as a general rule for 360 stuff if you want to see if it is good check out how well it details shmups -- the 360 got a bunch of Japan only stuff that is really quite notable and many things miss it) but I far sooner trust the old Scene releases to be complete there as they were all about getting everything they can. The 360 scene as a whole was also quite exhaustive in getting things; unless it was a promo released only for a hot minute or a title update doing said same (see something like the Fable Halloween update) then they would have had it; they also spent untold sums systematically getting all the music DLCs.
You also have the option to search for things in case Chevrolet decided to pull a Sega vs Shining games on youtube a few years back and memory hole it.
https://www.lumendatabase.org/notices/search?page=2&sort_by=&term=chevrolet+sonic+xbox
Scanned through a bunch there. False positives (probably should do a more exclusive search) and haynes manuals (not sure who would want one of those past about 1995 but hey).
Going another way then we have the dat sets, outside of ABGX360 (different entity to the ABGX site linked earlier) then this was less prominent in the 360 scene than it was for the GBA and DS (and MAME obviously) but no-intro are good people.
https://datomatic.no-intro.org/index.php?page=download&s=57&op=dat
"Test Drive Unlimited 2 - Chevrolet Camaro Synergy Special Edition (World) (Addon)"
gets added to the list of chevrolet related things if we were collecting them (though
https://web.archive.org/web/20190919201311/http://www.abgx.net/xbox360_dlc_releases_alpha.txt mentions it along with a bunch of preorder from given retailer DLCs).
About the only place I would go next is any internal listings from the companies themselves if they can be generated; if a list of acceptable games, update levels to certain games or similar can be generated then there is that. Don't think anything like that really exists for the 360 but I do usually find it comes more in backwards compatible devices looking to whitelist things and prevent their backwards compatibility being used for piracy. The DSi/3ds whitelist of pre RSA DS games being a good example of this in action, indeed I believe No-Intro mentioned above decoded it when checking to see if there were any v1.1 or whatever games.
At this point I am left checking archive.org copies of the main listing page
https://web.archive.org/web/2012011...c-Kinect/66acd000-77fe-1000-9115-d8025857099e
Seems a gamer picture got lost at one point. No mention of a game though.
As that is 2012 rather than 2011 (the Aveo coming out in NA as the Sonic in the same year, though marketing would have potentially happened beforehand) then we could try to go back further to see if the URLs changed schema before then (it would be around the time a few things got retools but I am also seeing other listings back in 2011 for other things there) and a game was lost. However at this point no mention on car or game enthusiast sites (beyond the free themes being noted) with several other mentions, no mentions on game listers that aim to be complete, no mention on piracy sites that also aimed for completion and were were within the systematic grabbing every piece of game content that was available (they spent a lot of time and effort making sure to grab all the music game individual track DLC, so much so I used to filter it out of my release posts, and as far as I am aware they paid for all those to grab them as well as unlike XBLA stuff there was no demo) regime; this was no more wild west PS1 days where everybody was doing their own thing from the local rental shop with their own preferred iso format.
Just in case someone thinks it might be one of those "only list the full thing if logged in" then seems not
https://web.archive.org/web/2012011...a-Kinect/66acd000-77fe-1000-9115-d80258410b79 or indeed for the 2011 thing
https://web.archive.org/web/2011062...a-Kinect/66acd000-77fe-1000-9115-d80258410b79
It could have been some tech demo that leaked out the back door a la the Goldeneye stuff and a few things besides
However at this point I am thinking it was nothing more than a few virtual posters the marketing droids at car company for the kids to aspire to (suppose it technically would be a hot hatch so something kids might like) and them to tick on their "beep boop I am down the the kids" list. However if pirates, gamers, gearheads, kinect fanboys and content starved sites for all those, all of which were routinely plumbing the depths of everything in full autist mode (and 2011 being well within currently accessible websites and extensively archived aspects of the internet) for scraps of info for decades by this point never thought to grab anything, and we have contemporary examples of that sort of enthusiasm for things from the same company, then yeah I think I am going with it was never more than a few themes from the marketing slimes of a company.