In the British monarchy, the monarch (“the Crown”) and the person who is the current monarch are considered distinct “people” with their own separate possessions (i.e. King Charles as the Crown owns property separately to Charles Windsor the private citizen).

So these oaths are meant to be pledging loyalty to the Crown, in its role as the embodiment of the British state, as opposed to the king personally.

The commons library is a treasure trove of information about the UK’s fascinating and complex constitution, I’d strongly recommend giving it a read if you’re interested in this sort of stuff!

Commons Library: the Crown and the constitution

In particular, I’d recommend checking out The United Kingdom constitution – a mapping exercise, which is a document intended to be a reasonably thorough summary of the UK’s constitution and where it comes from. It’s ~300 pages so I wouldn’t recommend reading the whole thing, but it’s great as a reference for the parts you find interesting.