Does Star Crossed require to have a friendly unit to be able to be played?

Yes, you must control a friendly unit on the board to be able to play Star-Crossed.

According to RiftJudge FAQ #9865 and #9452, Star-Crossed creates a targeting relationship with both a friendly unit and an enemy unit. Because it requires multiple targets without "up to" language, both targets must be present and valid at the time the spell is announced and placed on the chain. If you do not have a friendly unit on the board, the targeting requirement cannot be met, and the spell cannot be played.