He would be a prince if he'd popped at the capital; satellite cities of royal sides pop nobles. If I m remembering correctly.
I'm pretty sure that originated in the forums and has never been confirmed. It's interesting speculation and it's possible that royal units must pop in the capital, but if that is true then I would expect it to mean that the units that pop in other cities don't consider the ruler their father or mother. We've never seen a clear explanation of what makes a father or mother and I wish that would be explained, but until it is I'm going to strongly suspect that if a unit's father or mother is royal then the unit must also be royal. It would be strange even by Erfworld standards if Crush weren't a prince while the queen is his mother. What more does someone need to qualify for the title of Prince?
On the other hand, it's strange that Prince Axe is more than once called "the Prince," as if he were the only prince around.
This information is given in the Hardback edition for book 1. Said book has a glossary. I don't have it on hand right now but I'll try and remember to get it out when I get home and check the exact definition. But as I recall it say that Royals pop in the capital of a royal side and other cities controlled by a royal side pop noble warlords.