Spuff,
A number of Mexican UPU specimen stamps issued to Portuguese colonies have been examined by the EC, I have attached an example with the Ultramar overprint. Apparently the archive from one of the Portuguese colonies (I heard Gamboa) came on the market over the last few years All the examples we have seen are singles attached to manila paper with various overprints, Ultramar, Especimen, Specimen, Colonias, Amostra and Muestra.
However the UPU was formed after the Eagles were in use. The UPU founding was at the first congress held in 1874. Mexico became a member on April 1, 1879. I believe the first Mexican stamps distributed were the 1882 Juarez issue.
Regarding why multiples, stamps in blocks and sheets are sent to the Swiss who administer the distribution of stamps to members. The UPU preferred stamps be sent without overprints but that was not always followed. When received by members their own overprints were typically applied.
I can't imagine your Eagles were overprinted for UPU use, it is possible the overprint is bogus, or applied by some other entity for an unknown reason.
Marc