Robertson replaces Trent Alexander-Arnold as vice-captain behind skipper Virgil van Dijk after the latter's departure for Real Madrid.
Slot's decision affirms Robertson's future with Liverpool, after speculation he could be on the move this summer.
"It's Andy Robertson," Slot confirmed. "He's played here so many years, won the league twice. It was Virgil, Trent, Robbo, Mo last season (in the pecking order), and Trent left, so there are logical reasons for that.
"Apart from that, why is he the second captain? Because he's a very good player, he plays many games, he knows what it takes, but also he knows the culture of the club and the dressing room.
"Apart from all the quality he's shown on the pitch, he also plays a big role in the culture we have here at this club which - I've said this many times last season - is the biggest gift I could get from Jurgen.
"And that's also something we definitely have to keep for my successor to have, if I'm ever leaving this club."