Certain skills interfere with each other. For instance playing chess makes you worse at go, and playing go makes you worse at chess. Certain pairs of languages are likewise hard to learn together - my son found that he could not study both Russian and Chinese at the same time.
But in general you just develop more and better memories.
I spent years living in Taipei studying traditional Chinese before I moved to Moscow to study Russian. I actually found that they were linguistically distinct enough that it was easy to compartmentalize each language in my head without any cross bleed like you would have if you were studying Spanish and Portuguese simultaneously.
Yes, if you master one, then there is no conflict. But my son was a monolingual person learning another language. And he kept trying to apply Russian ideas to Chinese, and Chinese ideas to Russian. Therefore, even though he personally wanted to know Russian more, he chose Chinese instead because it met a school requirement for a second language. (And they couldn't teach him Russian.)
Certain skills interfere with each other. For instance playing chess makes you worse at go, and playing go makes you worse at chess. Certain pairs of languages are likewise hard to learn together - my son found that he could not study both Russian and Chinese at the same time.
But in general you just develop more and better memories.