As Michael said, crazy as it is, this is the rule, and there's no workaround. It's the confusing distinction between "authoriziation" and "association":
Most people are unaware of this until, like you, they get burned by it.
As always, I encourage everyone to write the angriest possible feedback to Apple about this stupid rule.
Thank you guys. After more research that's exactly what I found out. It's sad that I am prevented from buying iTunes Match. I understand that we can't share the same iPad or iPhone but an iMac or a MacBook Pro? Two different apple ids with different iTunes library should be able to share a computer and still use iTunes Match.
I wrote and email to Eddie Cue and Tim. Also submitted feature request on Apple support page and filed a complaint. Hopefully this will be worked out in the near future.
Like many of Apple's rules, this appears to be aimed at making illegal sharing just *a bit* more difficult. You can temporarily authorize your friend's computer to play your music, but you can't use Match to get the music there. Of course you can still use a thumb drive, hard disk, or generic cloud service. As always, it's trivial to break Apple encryption anyway, if that's your goal.
So this doesn't really limit illegal sharing, but can be a really inconvenient for legitimate, paying customers. In your case, they would only make money if you *bought a second computer* for your wife's content. As it is, they've lost even the $25/year Match fee.
Any concerns could be addressed by tightening up the rules on authorization (say, limit the number of times a single computer can be de-authorized and re-authorized), but maybe there are legal problems doing that after the fact (OK, I admit, I've never read the 60 page agreement when I buy stuff).
In a few years, we'll all have enough generic cloud space that we can just put all our music there and make this a non-issue. In the mean time, it's really annoying.