Generally speaking, man-made hybrids of ANIMALS are not given separate taxonomic status [apart from several millennia-old domestications]. Plants have special rules for it, but DO include many man-made species of hybrid origin.
Among animals, there are many recognized species of hybrid origin. In order to be recognized, the hybrids cannot be a first generation cross [ie, the parents cannot be different species - they must be part of the distinct hybrid gene pool]. Otherwise, the standards are basically the same as for any other [animal] species: they must maintain a gene pool in common with one another and distinct from any other species. Anything else is optional, but often preferable, such as distinctive behavior, habitat, habits, morphology, etc. As an example, most of the known parthenogenic species of lizard are of hybrid origin. They do not interbreed with either parental species and thus no longer share a gene pool. While they do not "breed" per se among themselves, they are clones and thus have more or less identical genomes. Frequently they are slightly different in morphology from both parents, and occupy an intermediate habitat.
In comparison, many toads hybridize, but do not result in new species. Anaxyrus americanus and A.hemiophrys hybridize over a broad band of habitat, but the hybrids interbreed freely with the parent species and do not restrict to themselves. Nor do the hybrid genomes integrate with the parent species - the 'foreign' genes do not spread in either parent population beyond the area in which hybridization takes place. That means that. The parent populations are distinct as species, the hybrids are not.
Axolotl-tiger hybrids COULD be considered a man-made species [which isn't done], but 1) the tiger salamander complex contains many species with only small and recent divergences anyway, 2) the hybrids have not been kept in any particularly unique gene pool - there is no distinction of what animals contain what proportion of genes from the two species, 3) the tiger traits for the most part have probably been swamped with the axolotl traits [that is, the tiger parts have mostly been bred out of existence, and 4) there is probably no way now to distinguish the captive introgressed specimens from pure wild axolotls.
Among other Ambystoma, the situation is more difficult. Among the kleptogens, apparently all forms share a single ancestral species which is now extinct but related to A.texanum and A.barbouri. These forms do not breed among themselves or true to themselves, but they also do not integrate the genes of the other hybridizing species - other species are required in order to breed, but they do not "mix". Instead, the kleptogens produce a couple of different kinds of offspring: offspring which have mixed genomes ("hybrids"), and those which are effectively pure members of one or more ancestral species. In the two main examples, the kleptogens often occur in places where one ancestral species does not occur, and they cannot produce that species as offspring. For instance, the kleptogens in Minnesota contain two sets of A.laterale genes and one set of A.jeffersonianum genes. They must breed with A.laterale to reproduce, but can only produce two kinds of offspring: pure A.laterale [laterale/laterale genome], and kleptogen offspring [laterale/laterale/jeffersonianum]. Pure A.jeffersonianum [jeffersonianum/jeffersonianum] does not occur in the state and cannot be produced by this form. In my opinion, these kleptogens should be considered at least one, and probably two distinct 'parasitic' species. While they must reproduce with other species, and can produce such as offspring, they have and maintain a separate range, distinct genome, unique ancestry, and unusual reproduction.
