There is definitely a PS1 emulator (as well as PSP and PS2 emulator) on the PS5 in order to play playstation classics (old PS1, PS2 and PSP games available for digital purchase on the playstation store). It just isn't unlocked for the user to do what they want with it for a ton of reasons I explained in another comment. Also the official Sony PSP emulator hasn't been worked on in a while, they only configure it to work for a few games, look at the sad state of these lists:
https://www.psdevwiki.com/ps4/PSP_Emulator_Compatibility_List
There is definitely a PS1 emulator (as well as PSP and PS2 emulator) on the PS5 in order to play playstation classics (old PS1, PS2 and PSP games available for digital purchase on the playstation store). It just isn't unlocked for the user to do what they want with it for a ton of reasons I explained in another comment. Also the official Sony PSP emulator hasn't been worked on in a while, they only configure it to work for a few games, look at the sad state of these lists:
https://www.psdevwiki.com/ps4/PSP_Emulator_Compatibility_List
https://www.psdevwiki.com/ps3/PSP_Emulator_Compatibility_List
Isn't the PS1 emulator actually running inside the PSP emulator?
Yes, it does on Vita.
The Vita system only does partial emulation of the PSP as there is a MIPS R4k CPU like the PSP auxiliary on the Vita system.
The PS3 system does not.
Not on the PS3 or PS4 as far as I'm aware, that could be the case for the vita