Vegas win the Stanley Cup the year prior with three goaltenders Im sure. So it’s not uncommon
True, but they had a bunch of goalies go down with injuries. I think I can comfortably say that most teams who have won the cup went into the playoffs with the strategy being to play the same guy in net all the way. Sometimes it gets derailed by a guy getting hurt, or getting lit up one game so you insert another guy the next game to settle him down. If it happens regularly you're probably just going to get eliminated instead of win the cup, but Vegas got away with it. Colorado the year before got away with it by having to play Francouz in place of Kuemper for a few games when he had that eye injury.
I do know of some examples where a team did not have that strategy entering the playoffs, but of the two I have in mind, neither won the cup. In 2003, Minnesota was generally alternating starts between Manny Fernandez and Dwayne Roloson. The Bruins swapped back and forth between Linus Ullmark and Jeremy Swayman recently.
I'll go back a bit for my own curiosity, so maybe I'll prove myself wrong as I do it, but I'm not sure if it'll make me budge from believing that the cup champion was at least trying to run the same goalie for the entirety of the playoffs, and only didn't because the plan got derailed.
Tampa Bay 2021: Andrei Vasilevskiy played all 23 playoff games.
Tampa Bay 2020: Andre Vasilevskiy played all 25 playoff games.
St. Louis 2019: Jordan Binnington started all 26 playoff games, but Jake Allen got an appearance in relief.
Washington 2018: Braden Holtby got 22 starts and a relief appearance. Philipp Grubauer got 2 starts (yanked in one of them).
Pittsburgh 2017: Marc-Andre Fleury had 15 starts (yanked in one). Matt Murray had 10 starts and one relief appearance.
Pittsburgh 2016: Matt Murray had 21 starts. Jeff Zatkoff had 2 starts. Marc-Andre Fleury had one start and one relief appearance.
Chicago 2015: Corey Crawford and Scott Darling had 19 and 4 starts respectively, and one relief appearance each.
Los Angeles 2014: Jonathan Quick started all 26 games. Martin Jones made two relief appearances.
Chicago 2013: Corey Crawford played all 23 games.
Los Angeles 2012: Jonathan Quick played all 20 games.
Boston 2011: Tim Thomas played all 25 games.
(the site I'm looking at no longer gives information on GP/starts, so it's GP only from here)
Chicago 2010: Antti Niemi 22, Cristobal Huet 1
Pittsburgh 2009: Marc-Andre Fleury 24, Mathieu Garon 1
Detroit 2008: Chris Osgood 19, Dominik Hasek 4
Anaheim 2007: Jean-Sebastian Giguere 18, Ilya Bryzgalov 5
Carolina 2006: Cam Ward 23, Martin Gerber 6 (this one does indicate Gerber as being 1-1 resulting in 4 non-decisions)
Nobody 2005: Nobody 0
Tampa Bay 2004: Nikolai Khabibulin 23, John Grahame 1 (this one does indicate that Grahame came in after Khabibulin got yanked)
New Jersey 2003: Martin Brodeur 24, Corey Schwab 2 (both in relief)
Detroit 2002: Dominik Hasek 23, Manny Legace 1 (relief appearance)
I didn't watch hockey prior to 01-02 so I'll stop. I think I can conclude that it's not common strategy. I can speak for 2007 in that I remember Bryzgalov getting into games because Giguere was unavailable due to a personal reason I don't remember, but Anaheim probably would have played him if he had been available.
Now, if you want to say that it might be a good strategy to try it, that's assuming you have two quality goaltenders, I would agree. Toronto this year with Stolarz and Woll is a good example.