I have an iPhone 5S running iOS 12.5.1. Is it possible to update to iOS 13.5?

No, the most recent version of iOS 12 is the end of the line for the iPhone 5S and iPhone 6. All phones newer than the 6 can be updated to iOS 14.x, but there is no way to update any phone to iOS 13.x.

iOS 12.5.1 is the highest possible for that model.

You have an iPhone 5. It can not be updated further.