MLB already tried to claim copyright over baseball statistics, an argument that doesn't hold much weight since facts are not subject to copyright.
Glenn Mitchell has an interesting article about the opposite situation - players claiming ownership under the right to publicity. The right to publicity generally arises from common law or state statute and comes up where, as Glenn puts it, the likeness is used "to create the false impression that the individual has endorsed its products or services."
In many ways, I consider this similar to the use of trademarks in metatags. The analysis is somewhat different because this claim arises under right of publicity and not federal trademark law, but ultimately both are used descriptively, and not as a designation of source.
As the Court discusses, the power to exclude others from using names and facts that are otherwise in the public domain is just too absolute given the nature of the use.