My iPad says it is up to date but it has only version 10.3.1 and I should be up to 13.1 I believe; help?

No 8 or 9 year old iPad can be updated to iPadOS 13.5.1. You most likely have iPad 4, which would be current at iOS 10.3.3 or 10.3.4 for a cellular model. If you need the advances of the newest iPadOS, you'll need a more modern iPad capable of running the newer iPadOS.

Usually when that happens then you have the latest OS installed that your iPad can run. Is it a 5th gen for sure?

In that case you are running the latest OS for that model iPad (4th Generation). A 5th Generation iPad would be able to run iPad OS 13.