@halloween20 Prince Phillip of the UK is
technically a prince-consort as a not-in-the-line-to-inherit-power spouse of a queen regnant, but he's generally
not referred to as such on account of him having been the prince of Greece and Denmark by birth, so he's still a prince in his own right...just not of England.
@Vincentius has it right: Were his marriage to suddenly end, Zenjirou would not be a prince in his own right (barring whatever honorary/courtesy titles he might have picked up).
On a related note, if a marriage of mixed social rank would barr the kids from inheriting the land and fancy hats, then the technical term for this sort of thing in IRL history is a
morganatic marriage, a marriage between two people of different social rank (e.g. royalty marrying nobility or nobility marrying a commoner) in which the spouse and kids can't get the succession rights. Succession laws varied with time and place, with succession rights sometimes overruled by specific new lawmaking...which were then sometimes ignored, leading to funtimes like the War of the Austrian Succession.
As an aside, I so need to catch up on this series. ^_^;