I haven't heard about CA & black bears specifically but many States do this for many species. They'll probably pull a pre-molar from the back side of the jaw & analyze for age, nutrition, etc. You'll never notice it in a taxidermy mount & unless you point it out on a skull mount, no one will be the wiser.
I agree with Hal. You wouldn't believe the information that they can gather from a tooth. Utah has been doing this for years on all the big game hunts except for the general season deer hunt unless you pull through a checking station and then they pull it for you. They require you to pull and tooth and send it in to them along with completing a survey on where and about the animal that you killed.
I've neverbeen asked to provide a tooth from any animal yet but would have no problem with it in the location pointed out. If the information is that valuable then it's worth it in the long run. The only teeth related information I've had experience with is when I was a kid in Wisconsin and you had to check in every deer. I remember they would pry the mouth open on every one of them to get an age estimate but that was all they did.
Again the more informatin we can provide hopefully the better they can manage the animals for us and our children.
i've used the teeth on horses to determine the age, and now that i'm more skilled at it, i can tell a little about the nutrition and health of the animal. once i determine the age, i can then tell what type of foods the horse has been eating (it is easiest to determine in younger horses, under 8 years old.)
learning about it was kind of fun, maybe i should learn a little bit about game animal teeth and apply the same knowlege to my hunting of the game animals.
I've done it several times as part of a mandatory check process for hunters. It's really easy with the right tools, and you won't notice it's missing. It's the first tiny little tooth behind the big canine (called a pre molar). Try to take it out without breaking the root. Do so by cutting the gun all the way around the tooth, and then pull very gently with pliers. You'll know when it feels loose enough to come out. If it breaks, then there's always the tooth on the other side.
The record books, that are put out by Boone & Crocket and Pope & Young and where I live is Henry Kelsey they are a great way to research a state or a province for big game. This does take some time but the knowledge it gives you, can be very valuable. Not all hunters have their trophies recorded, as where I live after the Hanson Buck became the new world record typical Whitetail, we saw a flood of hunters and now that has happened, some hunters are shying away from the record books but...