There are likely 2 separate answers. For apes, we share a close common ancestor, so we share many traits as well. In the case of specific faces with specific emotions, it’s all brain and neruon chemistry, similarly eviovedv communication, etc.
With animals with more distant common ancestor (and I can’t think of too many), it’s likely not actually that similar, but we “see” it as the same, e. g. baring teeth, furrowed brow, etc. It’s similar communication, but evolved down different branches of the evolutionary tree (a phenomenon known as “convergent” evolution, where similar traits evolved separately and weren’t in the common ancestor.)
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