If someone makes a fan species for a certain franchise, how should that be labeled? Like should the relevant franchise be added to the tags, with the species being "[species name]_(fan_species)"?

They would be treated the same as any other species invented by that same someone without any affiliation to a particular media. The fact that it's supposed to be a fan thing is not relevant information for our tagging style, and they would not receive an implication unless somehow made official.

If it's visually distinct/defined enough to separate it from its peers, it can be tagged like any other fictional species, like felkin or charr. The _(fan_species) suffix is unnecessary. It shouldn't be tagged with a franchise unless official elements of the franchise are visible (e.g. a cutie_mark for my_little_pony), a fan species alone isn't officially part of a franchise, without more to it.