Former Everton, Rangers and Scotland defender Weir, 43, was appointed as Danny Wilson's permanent replacement on Tuesday after spending two years on the coaching staff at Goodison Park under David Moyes.
When asked if Weir could emulate Rodgers and Mackay, Winter, chief executive at Watford for two and a half years before leaving Vicarage Road in 2011, said: "I hope so. They're very similar. There's an inherent determination in these guys that will see them through.
"There's a big similarity in that two of them are Scottish and centre-halves, which is another thing.
"They're very professional, very methodical, very well thought-out and clear about where they want to take things and very comfortable with the club's structural change that's been happening."
Winter returned to Bramall Lane last month for a second spell in the role as part of a restructuring of the board as the club works towards becoming self-sustainable.
He added: "I've worked in the past with two first-time managers. I worked at another football club and both Brendan Rodgers and Malky Mackay were first-time managers.
"People often see first-time managers as a greater risk than any other appointment, but I don't necessarily subscribe to that.
"Every manager needs the opportunity to prove himself, whether you're a first-time manager or not is pretty irrelevant, it's the skills, the qualities, the philosophy, the leadership style that they bring and you have a confidence that they can come in and do the job that needs to be done.
"No disrespect to any of the other candidates at all, because we had some really strong candidates, but we felt David was the ideal person to take the club forward the way the club wants to run."