Hey PetrCzech89,
If we understand correctly, whenever you restart your Mac the trackpad setting to wipe with two fingers is automatically turned off. If that's right, do you have any accessibility features for your trackpad turned on? If so, turn those off to see if that helps. Here's more information, just in case: Change Pointer Control settings for accessibility on Mac - Apple Support
If that's not the case, let's help to isolate this a bit more by creating a test user account to see if the same behavior happens there. This will let us know if the behavior is only within your user account or system-wide. Be sure the account is an administrator account so there's full access for testing: Change Users & Groups settings on Mac - Apple Support
Keep us posted on the results.
Take care!