MacOS – Does Time Machine need a “clean drive”, or can I use an old hard disk

Time Machine allows you to backup multiple macs to one drive. You just plug it in, set it up, and you're good. If you look at the file system, you can see separate backups for both computers. At our home we currently do this with a Mac Mini and a MacBook. The Mac Mini is always plugged in, and the MacBook gets updated every couple weeks. We use a 1.5 TB drive from Western Digital. Given your description, I think this solution with your 2TB drive should work well for you.

The data backed up by both macs will not be the same, as no matter how hard you try, both computers will wind up having subtile differences, so the data won't be "exactly" the same. Backing up only one system doesn't seem ideal to me, but you can restore a MacBook from a Mac Mini backup. It works fine.

Also, you realize that the drive must be connected for you to access the photos and music that you have stored there. You'll have to factor that into your storage solution.

Time Machine requires an HFS+ partition on the drive, it can't use an NTFS formatted one, even with MacFUSE installed. The way to do this would be to shrink the NTFS partition, then make a new HFS+ formatted partition to use with Time Machine (you can do this with Disk Utility). Unfortunately OS X can't resize NTFS partitions, you'll need to use a Windows machine to do it, then use OS X to create a new HFS+ partition.