A Few Sports that tolerances are not as good.
Baseball/Softball and Rounders is the two sports where say a bat can be different from bat to bat if using a traditional non composite wood bat simply due to using a once live object that growing conditions have an effect on.
Ice Hockey if using all wood traditional sticks they can feel slightly different from run to run for the same reasons that Baseball/Softball and Rounders traditional wood bats do though for hockey sticks the effect is not as bad as the three Bat stick sports.
Tennis the racket was once wood, same for Squash and they had variance in the wood same reasons as the other sports and just as much if not more for that then Ice Hockey, but since the 1980's when aluminum was then used for the better rackets the tolerance for a racket went way down.
these are the most played sports that use or did use wood for the equipment, not sure about how a Cricket bat is made but it could be wood too and that would make it so the bat has tolerances as well. Cricket is a highly played sport due to India/Sri Lanka playing it.
Now in most other sports the tolerance for the equipment is high enough that switching does not have much effect on a player unless that company dose not make a style similar to the equipment they were using. Even sports that use plastic equipment like some hocky sticks.
In the wood used for those sports If using the more traditional type however, you will have the bat or stick in ice hockey feeling the same due to most of them being machine made with strict tolerances, what is different is how the ball reacts to the bat or stick in hockey, some will have more spring then others and then how the bat or stick in hockey swings in the air, some might swing odd due to how stiff or how flexible the wood is simply due to how it grew.