i have an unlocked galaxy note 20 ultra 5g. prior to that i had a note 10+ unlocked. both 512gb units. i ave never had the issue you descorbe. your problem is because your app is either optimized, you are using a battery saving profile, or both. the note series has more options than the one plus and if this is your first time with the note series, its easy to miss some settings.
if you have ANY battery saver options turned on, the os will shut down background apps.
also check that the tesla app isn't "optimized" . go to settings - apps - press the 3 dots upper right and select special access. from there select "optimize battery usage". in this screen you will see apps that are andarent optimized. make sure the tesla app isn't optimized.
although i dont need to do this as i have my phone set to max performance, you can also go into settings/battery/app power management/apps that wont be put to sleep. you will also see "sleeping apps" and "deep sleeping apps". after some time has passed since you used your tesla app, check to see if its being pt to sleep there.