'Thing' can be used to mean an object, animal, quality, event, situation, etc. https://learnersdictionary.com/definition/thing In this case, 'thing' refers to the quality or aspect of his/her character.

Thank you very much.

Yes, you are correct. From the context of the conversation, the main character is being a bit flippant (maybe even snarky). As Michael said, the line could have been "I'm like that" or even "It's kinda my thing." would fit well within the context. (By the way, "kinda" is obviously not technically a word. However, it is used colloquially quite a bit and fits her demeanor. You would only use it in very informal situations.)

in addition to Michael's options, "It's a characteristic of mine" or "it's one of my characteristics". But Michael's "I'm like that" is easier and more natural.

Yes it does mean a characteristic in this case, and yes its use in this way is quite modern but is now widespread. "Thing" is a word with many uses :) Still, you wouldn't really use a "characteristic about me." You might say "I'm like that."