Here's the original French, target phrase in bold:
Bien sûr. Quand tu trouves un diamant qui n’est à personne, il est à toi. Quand tu trouves une île qui n’est à personne, elle est à toi. Quand tu as une idée le premier, tu la fais breveter : elle est à toi. Et moi je possède les étoiles, puisque jamais personne avant moi n’a songé à les posséder.
The word-for-word translation is simply, And me, I possess the stars...
. I would agree with your first interpretation:
So it is with me
Or if you prefer a clunkier, more more lengthy translation:
"And [so|as such] it follows that with me, I own the stars..."
This is due to the the logic that links the prior sentences to the following sentence. English doesn't use an "And me" emphasis, so I'm just replacing that with a somewhat different emphasis in English that follows the meaning. I think that's basically what the original translator did. They created their own stylistic emphasis that made logical sense.
@MishaLavrov points out that "as with me" tends to be more contrastive. I'd say the same about "In my case". One might think of "In my case" as being either neutral or contrastive, but the sentiment of the target sentence is comparative/similar. As such, the meaning inferred here ("So it is with me") is at a higher logical and relationship level, and not at a word level.
In conclusion, my advice to you would be to focus more on the overall logic of what is being said rather than focusing on the meaning of the single word "so".