If we are talking about limited-time Community Events, you can't check for an individual car's time in those on PC either. I'm pretty sure it only stores the best time.
On PC in the standard Time Trial mode you can view your best with each specific vehicle or by "classes" which are just broad categories like all GT cars or all Formula cars and don't relate to any of the actual classes in the game. But there's no way to jump to your own place in the rankings and see who's around you(and download suitable ghosts to chase) except to flip one page at a time. Not a problem when there's only 50 or 60 times posted, rather annoying if there's 600.
In other game modes - practice/quick race/career - the game does track your best times with each car but there's no way to view the times except on the loading screens. Only Time Trial mode times go to the TT leaderboards.
So the time recording feature is a bit lacking all around, but it is truly bizarre that they took out the vehicle filter on the leaderboards for console. That's such a basic feature, though it seems to be often omitted as I recall GRID and GT games, probably others. I think I remember reading about Sony/Microsoft being jerks about doing those because it is a major expense and connectivity issue or something. They probably just want to charge more for it because they can.
There is
http://cars2-stats-steam.wmdportal.com/ which is a bit more functional for PC but you still can't search for a user. There is also
https://projectcalc.org which attempts to combine the platforms(and does appear to have search). I think you need to download an app
https://projectcalc.org/download.php first to upload your console times to it which means there won't be very many console times, PC times seem to be included automatically since I found myself in the same 3rd position on Azure Coast in the 918 Spyder as the official board and I've never downloaded anything.
