Hi @BL4CK$K1LL
Well it worked but I think not because of some files being re-downloaded, but because I was starting the game from my second Steam account which I only use for CSGO.
Back in the days I accidentally redeemed the game key for ATS on my second account, while I wanted it to be on my main acc.
Last couple of days I started TruckersMP with my Main account which actually doesn't own ATS, but is gained access through Steam Family Share.
I can login at the TMP login screen normally while on my second steam account (which indeed owns ATS), but logging in while on my main steam account gives me the error again.
Conclusion: Just stupidity of me. Thanks for your help @BL4CK$K1LL and @Grundii!