So I'm a square! Well, at least I'm not a square peg in a round hole.

You had better be a round peg in a square hole than a round peg in a round hole. The latter is in for life, while the first is only an indeterminate sentence.

If a square peg doesn’t fit a round hole, neither the peg nor the hole is to blame. Between two people, the question "whose fault is it?" is the friend of argumentation and the destroyer of growth-oriented communication. Assigning blame involves listening to criticize and responding to defend, speaking to lower the other person rather than speaking to build each up. Relational progress is impossible as long as blame is the focus because blame and progress are enemies. In our litigation-hungry society we must take care that focusing on fault - which is proper for the courtroom - doesn’t carry over into interpersonal relationships. The heart of loving communication is listening to understand.

That's me. I can be me a bit at home, but I'm kind of like a square peg in a round hole.

This whole business feels kind of intense, like a bad fit. Round peg, square hole. But whatever, I'll take it.

They say "the taste of the pudding is in the eating," but that doesn't apply to "a square peg in a round hole".
If the peg is driven by hunger to give and desire to fill, and the hole on the other hand is moved by thirst to receive and purpose to fulfil, then a deep knowledge of the reason for existence, a mutual understanding of roles, and the wisdom to effectively carry out those roles are very key, if we must have a round peg in a round hole, or a square peg in a square hole.
But then again, who cares about "shape" in desperation?!