Interbreeding is not always a great way to figure out if animals are different species or not. I have just interbred two species of Devario fish, one from Assam, one from some Himalayan foothills.
It became popular for a while (IMO) because it was easy to explain and made sense from a genetic isolation point of view. However, there have always been situations where it did not seem to work well with the facts and people's ideas of what different animals were in different species.
There are variants on this idea involving breeding in the wild or breeding under certain circumstances, but they have their problems also.
For example, only a species if not breeding to some extent in the wild: What if a physical barrier to interaction between two potentially interbreeding populations (populations that could interbreed if they were able to come in contact with each other) arose or went away. Does this physical change in the environment cause the two populations to become the same species or different species? Does not make a lot of sense for classification purposes.
Another possible example
@Ygggdrasil mentioned is the ability of populations in the same evolutionary lineage, but separated in time, to breed (if they were able to get together). This is theoretically possible now that sperm can be frozen and stored for extended periods (or with seed banks). As one species evolves through time, earlier members can change (genetically or morphologically) which can result in biological barriers to breeding of various kinds. When does this reach the level of being different species.
Not only can domesticated dogs breed with wolves, but other canids can breed with each other (wolves coyotes for example), so the issue is not restricted to just dogs/wolves.
Cross species breeding is known in many situations with different effects on species naming.
Some
fish species have an extensive history of interbreeding which has generated a
reticulate (branching both apart and together) pattern of evolution.
Some
sunflowers form different species (in an isolated breeding population sense of the word) when they interbreed because they become polyploid and therefore genetically incompatible with either of their parental populations. They can only breed with other polyploids.
Some
insects become separate breeding populations when infected with particular parasites which make them incompatible with uninfected hosts. This can be cured by antibiotics. What is that?
In the last case, DNA similarity would not be very revealing since the DNA would have little time to evolve any differences.
DNA can be useful to rule out organisms being in the same species however (very different DNA, probably not the same species)
There have been morphology based definitions (species all look alike, which is probably out of date, but an initial clue).
There are also other more vague definitions like they evolve together as a group and share a common evolutionary history (which makes sense from a theory point of view, but is kind of hard to use as a functional definition in actual experiments or observations).
Probably the best advise would be to define your terms and use them consistently.
Here is a
wikipedia article that describes several different species concepts based on quite different ideas.
