Technically, since Pureblood got removed from the Yuan-Ti PC racial option POST Monsters of The Multiverse, your Yuan-Ti PC isn't a Pureblood right off the bat unless the player states he/she is. So yes, your Yuan-Ti pc can be/is a fully Snake Person. And...............
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If you click on this link, you will see that the Yuan-Ti Rogue pc Miniature, does indeed have the "Snake Bottom" as the legless Snake Tail that is popular with Serpent Folk imagery. As well as a Cobra hood. Forked Tongue would, I believe, be inside the mouth as per Nature's rules.
The lineage stat block also doesn't have any features reflecting challenges and advantages to having a snake lower body (unlike, say, the Centaur lineage stats). Really, it's more like the Tiefling's tail - if you want to fluff your character as having a Lillith-type body rather than a humanoid body, you can, but there's no mechanics there to support the fiction.