Yesterday I did an internet search for "arcade platform game change character in phonebooth".
The 3rd result was a reddit thread entitled "An old arcade platformer in which you could go into phone booths to change into different characters".
I felt a little moment of joy in that someone else had the same nostalgia quest as I.
(The game was Psychic 5, and I remember playing it as a kid at the corner store in Wagga near my grandparents house, but have never known the name)

