Adueñas comes from the verb adueñarse, meaning "to take ownership of" or "to make something one's own." It's a less common verb than simply saying "you have me."
In this song, the singer uses it to express how completely captivated he is by his beloved. He sings, "con solo un beso te adueñas, de mi" (with just one kiss, you take ownership of me), conveying a powerful sense of surrender and devotion. It highlights the profound impact her affection has on him.
Sergio Vega spends the first half of this song making promises he can't keep and the second half breaking every one of them on purpose. He opens as a suitor with grand plans: "Voy a bendecir tu nombre" ("I am going to bless your name"), roses, a small kept chest of dreams for her, shouting his love "a los cuatro vientos" (to the four winds). Then one kiss changes the terms completely: "Pues sabes que con solo un beso te adueñas, de mi" ("Because you know that with just one kiss you take me over").
From there the song stops being romantic and starts being funny. He lists what he's become, one job title at a time: "Soy tu esclavo... Soy tu mandadero... Soy tu gato, soy tu pistolero" ("I am your slave... I am your errand boy... I am your cat, I am your gunman"). A dog that follows her steps, a beggar living off one kiss, he runs through the whole résumé of devotion before landing on the title line: "Solo por ser el dueño de ti" ("Just for being the owner of you"). The joke, and the point, is that owning her cost him being anyone at all.