Why do some bathroom doors have round keyholes? Why don’t we see these types of locks on other doors? Are there round keys to fit in these keyholes?
Do you mean the ones on the outside of the stalls? If so, they can be opened with a tool like an Allen key (hex key) which the building caretaker will have, in case somebody gets locked in a stall and needs rescuing.
If you mean a domestic bathroom, I’ve never seen a round keyhole on one of those. 
Not sure if this is what you’re talking about, but on the bathroom doors in my house there’s small hole which you can insert a nail or paperclip or something into and unlock the door if it gets locked from the inside.
If you mean the ones with a key that’s a star-cross-section bar, then they’re not true locks, but simply a bolt set into the inside of the door - security isn’t an issue with locking a bathroom, and (as has been pointed out) being able to open the door from both sides if necessary is preferable.
why do some bathroom doors have round keyoles?
So your round eye can look in?
(and what’s a keyole?)
I assume you are talking about bathrooms you find in a typical residence. The little hole allows you to insert a small screwdriver (normally supplied with the lock) so that you can unlock it from the outside in an emergency.
Door hardware is available in three basic types: Entry, privacy, and passage. Passage sets can’t be locked-they permit passage at all times. Privacy sets are typically fitted to bedrooms and bathrooms, and are locked from the inside with either a thumbturn or pushbutton. An emergency opening is provided on the exterior lever or knob, and permits unlock of the fixture without breakage. Entry sets require a key on one or both sides, and are available in a dozen different functions, depending on the application.