They stayed that way: he/him/his playing his guitar, she/her singing her songs
First, the one palpable fault is the full colon, which is non-standard, where a comma would be expected.
Second, playing and singing are participles, that is, adjectives qualifying heand she, respectively.
And third, the subject form of the personal pronouns (he/she) should be used rather than the object form (him/her) or the possessive form (his/her[s]).
What has happened in this sentence is that the two agents (the man and the woman) are combined in the main verb and then separated into two participle phrases. Had the couple been (say) been just singing together, no pronoun would have been needed. You would just have
They stayed that way, singing their songs.
But there is a ‘but’. The preference for subject form has, I think, started to slip in casual, and even some not so casual, discourse. You will hear and even read more and more examples of the use of him/her where school teachers would mark them wrong. The most common is
Knock knock. “Who’s there?” - “Me”
I do not think there is anyone left who would answer:
It is I!