Why can’t persian relearn Pay Day after it’s evolved? Why does drowzee/hypno have Assist as an egg move? (the japanese name for the move is a cat-related idiom, isn’t it?)

1) I suppose because Pay Day is so closely associated in the designers’ minds with Meowth’s habit of collecting coins and other shiny objects (which Persian doesn’t share), and also with the basis of Meowth’s design in those little waving cat statuettes you see in Japanese shops sometimes (again, a feature Persian doesn’t share), which are supposed to bring luck and prosperity.  Discounting the existence of a first-generation TM, it’s actually one of the most exclusive moves in the game – the only other Pokémon who can learn it is Purrloin (and even then only as an egg move, which means that she can’t relearn it either, whether before or after evolving [EDIT: WRONG; as of X and Y you actually can re-learn egg moves]).  Contrasting that with Game Freak’s willingness to splash around other things that were originally signature moves, like Leaf Blade and Waterfall, it seems like Pay Day’s ability to generate wealth must be specific to the design of Meowth and Meowth alone.  Apparently this is really important to them.

2) They’re hardly the only non-cat Pokémon who learn it that way – so do Sneasel, Chimchar, Sentret, and Spinda.  To me it makes perfect sense that a manipulative Pokémon like Drowzee would be able to learn a skill that makes use of allies’ powers in place of his own.