iTunes home sharing

For Home Sharing to work the computer that holds the shared library, must be turned on. Otherwise, there's no way to access the shared Library.

However, yes, he can download items from a home shared library and import them into his own library. In which case those items no longer need to be accessed through home sharing but can instead be accessed directly in the recipient library without requiring the other computer be turned on.

See here: Import media using Home Sharing in iTunes on PC - Apple Support

The steps are in the link provded above. Here it is again: Import media using Home Sharing in iTunes on PC - Apple Support

1. In the iTunes app on your PC, choose a shared library from the pop-up menu at the top left.
2. Click the Show pop-up menu at the bottom of the iTunes window, then choose “Items not in my library.”
3. Select the items you want to import, then click Import.

You should also be able to right click on an item and then on Import on the context menu.

Family Sharing and Home Sharing are different services. See If you don't see your family's shared content - Apple Support for more on Family Sharing issues. Thinking about it the reason you may not have seen the import options might be that you are using simple network sharing rather than Home Sharing, where both computers would need to have been signed into the same Apple ID.

tt2

Sounds like you were trying to share Apple Music subscription music, and not purchases.

Songs and albums from the subscription service are not shareable over Home Sharing, you need Family Sharing and a Family Plan to the Apple Music subscription service for that and its access the way you describe.

2 different things, for 2 different purposes.

Make sure the computer is authorized for your Apple ID as well as being signed into it. I get Import and Add to Playlist > at the top of the pop menu you've shown.

tt2