Hart has received heavy criticism over his recent form, especially in the wake of Manchester City's Champions League defeat to Bayern Munich, with the England goalkeeper culpable for two of the European champions' three goals.
Former England defender Mills told Sky Sports News Radio that Hodgson was "exactly right" to give his number one a public vote of confidence on Thursday, despite calls for the 26-year-old to be dropped by club and country.
Hodgson's reassurance for Hart is in direct contrast to the approach former City boss Mancini employed last season, and Mills believes Hart is still struggling to regain his confidence, despite the Italian's dismissal in the summer.
Mills, a former team-mate of Hart, said: "I don't think the public criticism he got from Mancini last season has helped. That undermined his goalkeeper hugely and put a massive dent in Joe's confidence.
"I don't think it is right at all to drop him and Roy Hodgson has got that exactly right. It would be ludicrous to drop him as number one for these two games.
"I don't believe Manuel Pellegrini will drop him just yet. I think he still has a few mistakes (before that happens) and I'm not sure Costel Pantilimon is good enough long-term to replace Hart.
"Look what happened to David De Gea last year. He made lots of mistakes, came in and out of the side but Sir Alex Ferguson stuck with him. Now he is considered a great keeper.
"It's just a dip in form for Hart. He's made one or two mistakes that of course he doesn't want to make, but I'm sure he's going to bounce back."
Listen to Sky Sports News Radio's interview with Mills, in which the ex-City full-back gives an insight into his former team-mate's character and how City can help Hart get back to his best.