How do I restore a Mojave Time machine backup on Catalina?
Time Machine doesn’t operate differently when restoring.

If you restore a backup interval made on an older or same OS - it works.
If you attempt to restore a backup interval made from a newer OS than what is attempting the restore it will fail.

Since we don’t know what’s failing - you can either attach the Time Machine and indicate before you create the first user account you want to migrate. If you are past that, erase the Mac and then watch carefully in the setup steps where you are asked to choose to restore from backup.
If you don’t want to erase Catalina - open Migration Assistant and choose the data you want to restore from your Time Machine.
If you like steps and guides - here is a great one on moving your data to a new Mac.

https://support.apple.com/en-us/HT204350