stay vs stay on

stay is a noun but stay on is not a noun.

Word	Noun	Adjective	Verb	Adverb
stay	Yes	No	Yes	No
stay on	No	No	Yes	No

As verbs, stay on is a hypernym of stay; that is, stay on is a word with a broader meaning than stay:

Other hypernyms of stay include continue, remain, stay.

As verbs, stay and stay on are synonyms defined as:

Other synonyms of stay include continue, remain.

stay (noun)	stay on (noun)
continuing or remaining in a place or state
(nautical) brace consisting of a heavy rope or wire cable used as a support for a mast or spar
a thin strip of metal or bone that is used to stiffen a garment (e.g. a corset)
a judicial order forbidding some action until an event occurs or the order is lifted
the state of inactivity following an interruption

stay (verb)	stay on (verb)
stay the same; remain in a certain state	continue in a place, position, or situation
stop or halt
overcome or allay
fasten with stays
be in a certain place and not leave
stop a judicial process
dwell
continue in a place, position, or situation
hang on during a trial of endurance