"Lynx" can refer to either the Eurasian lynx (Lynx lynx) specifically, or to the genus Lynx and the four extant species in it (Eurasian lynx, Canada lynx, Iberian lynx, bobcat). And the game recognizes all the four lynx species as distinct animals if you use the full names. In general it understands imprecise common/genus names as hypernyms of the more precise species names, which is the correct way to do it IMO.
In general, of course, even distantly related animals may share a common name due to superficial similarities – what is "robin", for example? The American robin was named after the European robin by analogy, simply because both happen to have a red breast. The two species aren't even in the same family.
Likewise, it wouldn't accept “panther” because “tiger” was already there:
> I assume you mean “panther” in the general sense of any big cat.
Why on Earth would it assume mean that, of all things, rather than “black panther”? If it's gonna be pedantic about it, it could've complained about “leopard” and “jaguar” already being there (which they were) instead of complaining about an animal that nobody in their right mind would call a “panther”.
I believe the poster you're replying to understands that. They're noting that the complaint about panther was curiously because they had already listed tiger, which is practically never called a panther, and not because they already listed leopard, which is a cat that is often called a panther. The statement about meaning "any big cat" I would guess to be a confusion based on the name Pantherinae for the subfamily of Felidae of which all these big cats are part. Though the puma, which as you note is also called a panther, is in the different subfamily, Felinae.
I personally just tend to avoid the word panther, because it very often causes confusion as to which cat you're talking about.
In general, of course, even distantly related animals may share a common name due to superficial similarities – what is "robin", for example? The American robin was named after the European robin by analogy, simply because both happen to have a red breast. The two species aren't even in the same family.