Soy más que feliz.

why not “contento”/ “contenta”?

You don’t normally mix “contento” with “soy”, you use “estoy”. It’s a temporary state of mind.

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English Translation

I’m more than happy.

Hm, so why isn´t estar used here?

It depends on the meaning of “happy”. You would use estoy feliz / contento (I’m pleased) to describe your current state of mind, soy feliz if you are basically a happy person.