It's just joining the two clauses, but the subject 'I' is missing from the second: 'Each year has 365 days, and I wish your 365 beautiful dreams come true.' It's wishing that on every day of the year, the other person's dream will come true.

It is not actually correct English. When speaking, or writing quickly, it is fairly common to omit words accidentally. I agree with Matt T. that the word "I" has probably been left out.