There is no DMCA analogue for trademarks, so, for example, YouTube has no obligation to automate trademark complaints or to resolve disputes between trademark owners and video creators, so almost every case will go through the court, and overloading courts with bogus cases can have consequences for them.
YouTube technically has no obligation to comply with DMCA takedown notices either; there is no direct financial penalty for failing to do so. Rather, the DMCA provides a "safe harbor" exempting YouTube from liability for infringement, then removes that safe harbor if they fail to comply with a takedown notice.
For trademarks, there simply is no safe harbor, and thus no conditions under which it can be removed. That means there isn't a codified process for YouTube to follow; but the stick they can be beaten with at the end of the day -- infringement litigation -- is the same in either case. You can bet they'll do what they can do avoid it.
