This is a quirk of digital music and online shopping in that the license to listen to that particular bit of music, which you purchased, results in you getting a digital file with a restriction built in. Without that restriction, there would be nothing preventing you, or anyone else, from distributing that file to the rest of the internet world. If you have more than one Apple device, an iPhone or iPad next to your iMac for example, I would suggest looking into the Home Sharing, or Family Sharing option. A phone is always on anyway, and if you then set up home sharing on this phone so that your wife's ID can listen to your music, the end effect will be pretty much the same as what you were asking about. There appear to be some differences between the two, but one phone call to Apple support might be worthwhile as well, they can suggest the correct solution for your situation.
(see here for Home Sharing:
https://support.apple.com/en-us/HT202190, the family sharing link is in my previous post.)