Union defender Danilho Doekhi scored either side of an incredible solo effort from Luis Diaz. Kane's late equaliser was his 13th goal in 10 league games this season.
"Winning so many games in a row is obviously incredible, but sometimes it becomes the main talking point and it might motivate other teams a little more," Kane told Sky.
"We're glad it wasn't a loss and we move on."
Despite dropping points, Bayern sit six clear atop the table, with RB Leipzig and Borussia Dortmund also slipping up on Saturday.
"It's a really good point from us - we got a point and the competition didn't," said Bayern captain Manuel Neuer.
Bayern had won 16 from 16 in all competitions to start the season, a record in Europe's top-five leagues, but looked tired after fighting with 10 men for much of Tuesday's 2-1 win at Paris Saint-Germain.
Kompany made just one change to Bayern's starting XI on Saturday and it showed, with the visitors sluggish and struggling to create much in the early stages.
Union had an early opener chalked off for a tight offside decision but took the lead with 27 minutes gone, Doekhi sliding the ball under Neuer from close range.
The defending champions were pinned in their own area but scored on one of their first forays goalward, Diaz somehow conjuring an improbable goal before half-time.
The Colombia forward saved a ball from going out and beat a defender on the turn, before blasting in from a tight angle.
Doekhi knocked a rebound through a crowded penalty area and into the corner of the goal to put Union in front with seven minutes remaining.
But with Union on track for a first-ever win over Bayern, Kane scored in the 93rd minute to keep his side unbeaten this term.
Union coach Steffen Baumgart told reporters it was his side's best performance since he took over in January.
"That's how you need to perform to get something against Bayern," he said.