It's not a lion. Or strictly a tiger either, despite the Japanese name. It's, "what if we took the mythological lynx, and instead of basing it on the (comparatively lame) actual animal, we invented a fantasy big cat interpretation of a lynx". So they throw in vague elements of tigers and lions in there to make it feel like a big cat, but it's not actually meant to BE and lion/tiger. It's the lynx, but reimagined as a bigger more fantastical animal of its own kind.

It's also not a sphinx, not in the goddamn slightest. You just got confused by Shinx's name when you were 14 not knowing what a lynx was and refuse to let it go.