The protocol on royal titles is archaic and let's face it, no ordinary person knows any of it. You'd have to have a special interest or work for the palace to have a hope of getting it right.
They've moved to the US. The US doesn't have our rules on royal titles as they don't have a royal family.
They're no longer working royals. So, they're not bound by the rules in many ways. They're supposed to be ordinary people. Sort of.
Ordinary people change their names all the time by deed poll. When it comes to name changing, the concept is generally legal as long as people still know who you are.
They could change their names by deed poll to something utterly random and bizarre and everyone would still know who they were. So... it doesn't really matter.
And having Sussex as a surname sounds perfectly reasonable. Well, apart from the fact where they have left the UK and yet want to be named after a piece of the UK, but horses for courses. Maybe they just like the way it sounds - it's only two syllables, so quite neat.