My iPad Air says its software is up to date but I need to update to iOS 13.5.5

The first generation iPad Air, iPad mini2 and mini3 can only be updated to iOS 12.5.5.

Apple ended major update support for these models of iPad in September 2019 - although they since continue to receive “point” security updates - the most recent being released just recently. These devices cannot be updated to iPadOS 13 (or any later major versions of iPadOS) as the internal hardware does not meet the minimum technical requirements (CPU and RAM) for newer versions of iOS/iPadOS.

Some App developers have dropped support for iOS versions preceding iOS 13.x for technical reasons (often because APIs or software libraries upon which they rely have been updated in the new version of iPadOS, making their App incompatible with earlier versions). 

Whilst many App titles remain compatible with your iPad, if you want or need to access to Apps requiring the most recent versions of iOS, you’ll need to consider replacement of your iPad with a newer or current model that supports iPadOS15.

Sorry, I appreciate that this may not be what you were hoping to learn.

I hope this information provides clarity as to your available options - and the fundamental limitations of your iPad.

There is no such thing as iOS 13.5.5. Apple never released 13.5.5. They moved from 13.5.1 directly to 13.6.

Regardless, if your iPad is running iOS 12.5.5, then it's likely an iPad Air First generation and can cannot update beyond that as it's too old and lacks the hardware required to run a newer iOS version.