Aside from the fact that song lyrics often follow the flow of the singer rather than any grammar, there is no reason why any of those versions are not possible. The verb forms (tense and voice) usually depend more on each separate action: We are snuggled up like 2 birds would be snuggled up [if they wanted to snuggle up / be snuggled up]. We have snuggled up like 2 birds would be snuggled up [if they wanted to snuggle up / be snuggled up].
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