Well, if my Google searching is correct, the surprising answer is no. From what I can tell, Yamaha did not provide the registration feature on the E300 series until the PSR-E343.
The E400 series -- from the PSR-E403 on up -- always had registrations, but even so, there were only a measly 8 banks of 2 registrations, for a total of 16, until the PSR-E433, in which Yamaha finally included a more reasonable 8 banks of 4 registrations, for a total of 32. And here we are, in 2022, with the PSR-E473 available -- 4 generations past the E433 -- and still, only 8 banks of 4 registrations available!
Really, unless you're just looking for something to use as a MIDI sound source or something like that, I'd definitely chose a model with at least some registrations. With the ability to split the keyboard, layer the sounds, and add effects, you definitely want a way to store these settings for easy recall during live playing.