The article doesn’t say, but I suspect that Mexico isn’t claiming IP on the ancient art itself, but on the actual photograph picture created by its museums of that art, which Starbucks (or their supplier) presumably cut and pasted onto their mugs. The museums then use the royalties to help fund themselves. At least, that’s been the case in similar cases I’ve heard of.
Not sure how I feel about it, on the one hand, funding for preservation of artifacts is a good thing, and getting good pictures of the items is probably non-trivial in at least some cases. On the other hand, one would think countries like Mexico would want to encourage images of its cultural heritage, since it promotes tourism, culture, etc.
There’s actually precedent for this, even in the United States. Look up the legal protection that the US Olympic Committee enjoys over the commercial utilization of the word “Olympic,” for an easy example.
I’m guessing that these were Starbucks’ cups in Mexico, though, right? I can’t see the US interfering in Canada, for example, over the commercial use of “Olympic.”