If the Music app works like iTunes, you should be able to drag-and drop tracks out of the Music window, onto the Desktop or a folder, to make copies of them. Then you can import those files into Music on the other Mac.
As for getting them onto your iPhone, you can use "manual" Finder-based synchronization. Here you would hook up your iPhone to one of your Macs, and use Finder to load the iPhone with music from that Mac's Music library.
Use the Finder to sync your iPhone, iPad, or iPod touch with your Mac - Apple Support
You can sync "music, movies, TV shows, podcasts, audiobooks, books, photos, contacts, and calendars" this way. You don't have to synchronize "photos, contacts, and calendars" manually just to synchronize music that way; and I'd suggest synchronizing "photos, contacts, and calendars" through iCloud (for greater convenience).
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Other ways of getting songs between devices are by using the iTunes Match subscription service, or the similar Sync Library feature of the Apple Music subscription service.
My understanding is that when the Apple Music "Sync Library" feature downloads "matched" songs, they might contain DRM that will keep them from playing if you ever stop subscribing to Apple Music.
Subscribe to iTunes Match - Apple Support
Use Sync Library with your Apple Music subscription - Apple Support