Fish said: “It’s a huge club. The direction that the club want to go in, you’ve seen a lot of good signings in this window.
“Being a part of that is really exciting and I’m happy to be here. Hopefully I can do my best for the team.”
The arrival of Fish is part of a busy few days in the market for the Bluebirds, with the transfer window closing at 23:00 BST on Friday, 30 August.
Cardiff have agreed the transfer of centre-half Mark McGuinness to Championship rivals Luton Town in a deal which could eventually be worth £10m.
They have also signed former Norway Under-21s defender Jesper Daland from Belgian Pro League outfit Cercle Brugge, for a fee of about £3.5m.
Cardiff have conceded seven goals in their opening two Championship games of the season and, after the 5-0 defeat at Burnley last weekend, manager Erol Bulut said he was hopeful of making signings this week.
Bulut said of Fish’s signing: “He is a young defender, who has been at Manchester United for many years, so has had a good education from them.
“For the last two years he was in Hibernian, so we watched him and we were interested in him in January too.
“We hope that he will progress with us in a good way. He is still 21 years old, just starting his career, and he is good on the ball, good in duels, so he will add to our playing style.”
Fish is Cardiff’s seventh signing of the summer following Calum Chambers, Chris Willock, Alex Robertson, Wilfried Kanga, Anwar El Ghazi and Daland.
