There are several ways to understand Isa 23:15 and its song of the prostitute but the simplest is this - V16 is either the song or a paraphrase of the well-known song. This means, as Barnes notes:
Shall Tyre sing as an harlot - Margin, as the Hebrew, 'It shall be
unto Tyre as the song of an harlot.' That is, Tyre shall be restored
to its former state of prosperity and opulence; it shall be adorned
with the rich productions of other climes, and shall be happy and
joyful again. There are two ideas here; one that Tyre would be again
prosperous, and the other that she would sustain substantially the
same character as before.
That is, Tyre was characterized by political and moral harlotry before being subjugated by Babylon. Afterward it will recover its independence and return to that same position of political and moral harlotry and, again, sing the song of the prostitute.
Gill arrives at a similar conclusion:
after the end of seventy years shall Tyre sing as an harlot; being
rebuilt and restored to its former state; as a harlot who has been
cast off by her lovers, on account of some disease she has laboured
under, and through a dislike of her; but, having recovered her health,
makes use of her arts, and this among others, to sing a song, in order
to draw, by her melodious voice, her lovers to her again; and so Tyre
being built again, and out of the hands of its oppressors, and
restored to its former liberty, should make use of all arts and
methods to recover her trade, and draw merchants from all parts to her
again.