The former striker left Tottenham for Old Trafford in June 1997 and went on to win the Champions League and three Premier League titles with Sir Alex Ferguson's side.
This weekend, United returned to the top of the league with a 5-2 victory over Spurs and still have a shout of winning four trophies this campaign.
Sheringham said such a huge weight of expectation can be hard for players to deal with.
"I remember him (Ferguson) saying to me 'you won't realise what it is like playing for Manchester United' and being a young cockney lad who thought I knew it all; I was like 'bring it on, I'm ready for it'," said Sheringham, who scored 46 goals in 153 games for United.
"With Manchester United we would go to places like Leicester and not give them a kick of the ball all through the game.
"Yet if the score was 0-0 with five minutes to go the Leicester fans would be going crazy because they were close to taking a point off Manchester United.
"But if we went there with Tottenham, in the same scenario the whole fans would have turned on the home team to get forward and have a go at us.
"But against Manchester United those same fans are still happy that they are getting a point.
"That's the difference. You have to win every game with Manchester United - that's not the case with any other club in the country. Perhaps Chelsea are getting to that stage as well."
Sheringham scored 125 goals in 277 games for Spurs in two spells, from 1993 to 1997 and from 2001 to 2003.
"It was a dream come true for me to join Tottenham. It's a fantastic club to play for - the stadium is beautiful, it's a proper arena to play in.
"We had a couple of good sides along the way and we were close to being very good but we just weren't close enough and that was the disappointing thing for me.
"I got frustrated because we weren't quite getting the success. We were a nearly team for so long and that was very disappointing."
Sheringham felt he had no option but to make the switch to Old Trafford, but admitted that he did not find the transition between the clubs easy.
"I got a little bit of stick when I first went there," he recalled." In the first year I didn't win anything and that was rammed right down my throat by a number of supporters.
"My first fixture when I left Tottenham was Manchester United v Tottenham at White Hart Lane on the first day of the season. I got a bit of stick that day. I played ok but I missed a penalty - I hit the post."
Better times were to come, though, not least the 1999 Champions League final when Sheringham - with United goalkeeper Peter Schmeichel up in the box - cancelled out Bayern Munich's one-goal advantage in the first minute of added-on time before Ole Gunnar Solskj?struck the winner.
"That shows how late on it was in the game: you only chuck the goalkeeper up into the area for the last minutes to get a last-gasp goal," said Sheringham.
"We got the last-gasp goal and still scored another one after that; it was an amazing turnaround, an amazing feeling that we had won the treble."