Brown, who played under Mellon at Fleetwood in the Conference, was at Mansfield last season and turned down their offer of a new deal.
The 26-year-old left winger is happy to link up with Mellon again at the newly-promoted League One club.
“It wasn’t a tough decision to come here – I’ve known Micky for a considerable amount of years and the chance to come and play under him was a no-brainer really,” Brown told Shrewsbury’s official website.
“He brought some of the best football out in me, so with the lads here and the momentum after promotion it would have been daft to turn down an opportunity like this.
“Micky’s got his own playing style and he wants to play total football.
“He knows how to get the best out of me and at this stage of my career; it was an opportunity I couldn’t pass up.
“When I was in the side there we were known as a team that won by playing attractive football and we had success under him.
“I’ve seen the highs under him and this is an opportunity to come here and achieve more success.”
Brown has made over 200 Football League appearances for Fleetwood, Tranmere, Oxford and Mansfield.