The 26-year-old rejected a deal to stay at Port Vale after being their 12-goal top scorer in his only season – with Valiants defender Ryan McGivern also moving to Greenhous Meadow.
Leitch-Smith started his career at Crewe, where he netted 22 goals in 122 games from 2009-14.
Legendary Crewe manager Dario Gradi watched him come through the ranks at Alex and believes Town have acquired a similar player to Tottenham and England striker Kane.
"Harry Kane is a superstar and AJ's not as big or as good as him but he's more like him than Jamie Vardy," said Gradi.
"He's a Harry Kane – I don't want to burden him with that, but he's that kind of player.
"Harry Kane isn't a terrific player until he scores a goal and that's what AJ is. He's a good player who's lost his way a bit but has always been a goalscorer.
"He's got to play as a No.9. As long as they play him through the middle as a striker he'll do well."
"I used to say to him 'why do you spend so much time on the wings?' and he said 'because they want me to run about more' and I'd tell him to stay in the box.
"He's not lightning fast but he's mobile and is a decent shape. If he gets a chance, he'll score.
"I was sorry to see him go from Crewe but that's not my decision any more."
Gradi revealed Town beat off competition from several other clubs to land Leitch-Smith.
"I spoke to him two or three weeks ago and he mentioned quite a few clubs were in for him," he said.
"He mentioned Shrewsbury and that was certainly the most convenient club location-wise for him. I'm pleased he's gone there because it's a nice club. If I get a chance to watch him play, I'll pay a visit."
Leitch-Smith is 26 and has a modest record of 31 goals in 173 League games. But Gradi believes he could play at a higher level.
"I've known him since he was a kid. When he was at school he lived over the road from the ground and I regard him as a friend and I really like him," he added.
"I think he could go higher. So if Shrewsbury get promoted next season he'll be fine."
Gradi says Leitch-Smith is a good character and will fit in well with Micky Mellon's squad.