The first musicality is simply to see the poem as a song lyric. There is nothing simple about it, but just consider that, functionally, that is what the text is, in this particular case, that it doesn't have much existence out of that context. We could print the lyric and appreciate its craft, but we never forget that it is a song lyric.
The second musicality is the poem's prosody. This might be a secondary effect of the fact that it was written to be sung, or that it is written in a poetic form identical to that of song lyrics. Here, though, we see that text as musical in its own right, not simply because it happens to be a lyric. With this case, paradoxically, singing the text might override the subtle effects of the versification, running roughshod over the poet's art. Or, a good setting might respond to that prosody, bringing it out in subtle ways.
The third musicality is musicality as metaphor for poetics. Here we could distinguish between a merely conventional use of the metaphor, and a deeper poetics. The merely conventional reference to a poem as song-like, or the poet as singer, is simply a nod of the head to that musical connection, but may not involve anything deeper than that.