I have heard people saying "happy birthday" but "happiest birthday" sounds a little weird to me as it is the superlative form. Which would mean that only the recent birthday to be the happiest only, no? I'm confused :/ could someone please explain the logic behind this?
There is nothing wrong or weird about Happiest birthday. For example:
It was my happiest birthday as I received more presents than I expected.
Yesterday was the happiest birthday of my life.
I hope you will have the happiest birthday with your fiance in London.
As you mentioned, happiest is the superlative form of the adjective happy. It doesn't mean it is the most recent birthday. If your happiest birthday was 10 years ago, you could say, "10 years ago today was the happiest birthday of my life."
However, you don't usually say "Happiest birthday to you." when you wish someone a happy birthday.