Mutts are not hybrids; they are a species in its natural state, that is, outbred. Purebreds are not hybrids either, but they are the products of consanguineous matings, that is, they are inbred. In the aquarium hobby these kinds of strains are often called 'line bred.' As a geneticist, you know this, so my explanation is geared primarily to those who may be confused by reading our comments on this issue.
Interspecific hybrids are an entirely different matter, and they pose a real threat to the maintenance of bona fide species in captivity. I have no issue with hybrid fishes provided they stay in your fish room; the problem arises when they are distributed, either knowingly or unknowingly, to unwary aquarists. To appreciate the magnitude of this problem, one need only peruse the 'Unidentified Cichlids' section of this forum, where you will discover that the majority of these queries concern interspecific hybrids of uncertain parentage. The problem is so acute amongst Malawian fishes that it is getting more and more difficult to find bona fide species in the hobby.
The problem is potentially even worse, if that can be imagined, amongst very closely related populations, such as the Tropheus species flock of Lake Tanganyika. These populations evolved so recently that effective meiotic and/or behavioral constraints to interbreeding have not yet been established, so they interbreed readily in captivity. So far, serious Tropheus keepers have been keenly attuned to this issue, and generally keep each geographically distinct population in a 'species tank' (a misnomer in this context) so as to eliminate the problem of hybridization. The fact that Tropheus are much more demanding in captivity than say, Mbuna, has also served to largely preclude inadvertent hybridization by unwary and/or uninformed aquarists.
The American Cichlid Association has as its mandate the conservation of existing species, and the continuation of wild-type genetic populations so they are not lost forever through pollution, habitat destruction, over-fishing, or hybridization. I agree with that mission in its entirety, and as such, I would never keep interspecific hybrids.