Je ne suis pas si douée que ça

just means:

I’m not that talented.

It doesn’t. A literal translation of “Et puis, je ne suis pas si douée que ça.” could be “And then, I am not as talented as that”. “Pas si [adjective] que ça” is an idiom that means “not very [adjective]”. In English, you can express a similar degree of “not very” by “not that [adjective]”.

“Je suis douée” on its own means “I am talented” (in general), and “je suis douée pour …” means “I am talented at …”. In the context you’re quoting, it’s implied in French that the statement is qualified: it’s about the specific thing mentioned in the previous sentence (succeeding in that environment). In English the unqualified statement feels a bit broader, so it’s more natural to say “talented at it”. In French, you could say “je ne suis pas si douée que ça pour ça”, but it feels a bit awkward, and it’s also repetitive due to the double use of “ça”, so we would omit “pour ça”.

You can say “je suis douée avec ça”, but only to express talent with some specific tool or method. Here it’s about talent to achieve an objective, so the correct preposition is “pour”, not “avec”.

“Je ne suis pas si douée que ça” could means :

I’m not talented as much as you think.

Or

I’m not talented like others can be.

It’s generally the answer of the sentence: “You are talented!”

This means you compare yourself with the talent of others people and you think that you don’t have enough talent.