Is my iPhone SE (iOS 10.5) compatible with Apple Watch Series 10?

If it's the first generation iPhone SE, no. They are not compatible.

If it's actually running iOS 10.5, you really need to update it. That phone can be upgraded as far as iOS 15. 15 is the end of the line for that model.

That would mean it is a second or 3rd generation iPhone SE. Yes, if it's one of those and running 18.5, it is compatible.