The designation PBEPBE is an actual terrible artefact from researchers using Gaussian. The actual publication[1, 2] only refers to it as PBE, and most other program packages (I know) implement it as such. Often they make the correlation and exchange parts separately available via PBEC and PBEX, or similar. Unfortunately this designation also made it into the literature.
The confusion further continues since Gaussian uses the PBE1PBE designation for the (standalone) hybrid functional, which is in most other packages known as PBE0.
But it doesn't stop there, for TPSS you may find similar naming/ reference schemes. The BP86 functional keyword requests the VWN(III) version, while many other programs use VWN(V) for this. The latter is available via BVP86, but misses any implementation of dispersion. We could go on with this...
In the end what is important is the correct referencing of the methodology so that the results are reproducible, i.e. look at the literature cited. Gaussian actually does a good job referencing those sources. And when you publish, you'd do us all a favour and not use the keyword designations of Gaussian.
 John P. Perdew, Kieron Burke, and Matthias Ernzerhof, Phys. Rev. Lett. 1996, 77, 3865. DOI: 10.1103/PhysRevLett.77.3865. Mirrored at Burke's page: dft.uci.edu/publications.php
 (Erratum) John P. Perdew, Kieron Burke, Matthias Ernzerhof, Phys. Rev. Lett. 1997, 78, 1396. DOI: 10.1103/PhysRevLett.78.1396