Media playback is not supported on this device
The Foxes beat Hull 3-1 on Saturday and have won both of their games under Shakespeare since Ranieri's sacking.
"Suddenly he wants to be a manager. It doesn't sit that comfortably with me," Keown said on Match of the Day.
"If you're assistant manager to Ranieri and he's walked, why is it you suddenly want to be the manager?"
Shakespeare, 53, has never managed full-time and was brought to Leicester by Ranieri's predecessor Nigel Pearson.
He says he expects to speak with the Leicester hierarchy about his future next week.
"You have personal ambition, but I think it's almost out of order that he wants to jump in for that job," said Keown.
"Surely they'll go for a manager who is bigger with more experience."
Former Liverpool midfielder Danny Murphy added: "While Craig Shakespeare is winning, leave him in charge. Why change? Let him carry on."
Speaking after his side moved five points clear of the relegation zone, Shakespeare said: "My remit was to win these two games and that's what we've done," he said.
"The owners will make a decision for the good of the club and until I talk to them I don't know what that will be. But as I have said previously, I'm comfortable with that.
"I can't control it and there's no point worrying about it."
Leicester had not scored a league goal in 2017 until Shakespeare took charge, but have now scored six in two games - twice as many Premier League goals as they did in their last 10 matches under Ranieri.
Shakespeare has never managed a club before but has a long association with the Foxes.
The Englishman was assistant manager to Nigel Pearson for two years from 2008, following Pearson to Hull in 2010 before they both returned to Leicester a year later.
Following Pearson's sacking in 2015, Shakespeare remained at the club as Ranieri's assistant.
Leicester are believed to have spoken to a number of potential candidates to replace Ranieri, but could also consider giving Shakespeare the job until the end of the season.
The Foxes have held informal discussions with former England boss Roy Hodgson.
The 69-year-old has been out of work since leaving the international set-up after the Three Lions lost to Iceland at Euro 2016.
Pearson, who saved Leicester from relegation in 2014-15, has also been linked with a return.

A further 111 are thought to have been wounded in the strikes.
Armed Forces Minister Mike Penning revealed the figures after a written parliamentary question by Lib Dem leader Tim Farron.
He said they were an estimate because strike sites cannot be visited and assessed by the UK.
Mr Farron had asked the Ministry of Defence how many militants from the terrorist group had been killed by British forces in Iraq from 2 December 2015 to 2 December 2016.
Mr Penning replied: "During the period in question it is assessed that there were a total of 111 enemy wounded in action and a total of 1,306 enemy killed in action as a result of UK air strikes.
"The UK cannot visit strike sites and conduct detailed investigations on the ground in Iraq. Therefore the number of combatants killed and/or wounded is an estimated figure only."
The UK parliament backed British participation in air strikes against IS in Iraq back in September 2014. Just over a year later in 2015, MPs authorised air strikes against IS in Syria.
The UK has conducted more than 1,200 airstrikes in Iraq and Syria since it became involved - more than any other coalition country bar the United States.
BBC analysis has shown that in 2017, UK forces dropped bombs in Iraq and Syria on 69 of the first 99 days of the year.
RAF Typhoons and Reaper drones have been supporting Iraqi and Kurdish forces trying to liberate Mosul, northern Iraq, in recent months.
Leicester City caretaker boss Craig Shakespeare is "out of order" for wanting to replace Claudio Ranieri, says ex-Arsenal defender Martin Keown.
[NEXT_CONCEPT]
More than 1,300 militants from so-called Islamic State were killed by British air strikes in Iraq over a 12-month period, according to new figures.