Amazon seems to think so. Their US product page for the device says: "WARNING: California’s Proposition 65." (I'm not sure if that notice appears for all customers or just those in California). However, the Park Tool CB-4 chain cleaner solvent "does not contain any Proposition 65 chemicals," according to the data sheet.
But I, personally, would use it regardless, and do so. It's an innocuous plastic item that doesn't come remotely near your food, not markedly different from the plastics you touch every day without confirming whether they contain P65 listed chemicals. Do your handlebars (some grips contain lead) or shifters contain P65 chemicals? Do you inhale exhaust fumes while cycling? Do you eat baked, fried, or roasted foods? Touching the handle on a chain scrubber is just so far down my list of worries compared to all the other risks in the world.
Like Greg Hewgill said, I use gloves anyway with my Park Tool CM-5.3 Cyclone Bicycle Chain Scrubber, because chain cleaning is a bit messy, and I'd rather wear gloves than scrub muck off my hands later.
then the absence of the P65 warning is a mark of quality
I would not assume that at all. The no-name clothing manufacturer may have never heard of Prop 65, and the importer and retailer may have no idea whether or not the good should have a P65 warning. The presence of absence of the warning is really not correlated with risk of harm. Worrying about the risk of brief touches of a plastic item on an occasional basis is to hyper-focus on small details instead of broad environmental risks. And there's absolutely nothing to say that another brand of chain scrubber won't contain P65-listed chemicals but simply not have the warning. As that Wirecutter article notes, the fact the product comes with a Prop 65 warning at least means whoever put it there has at least minimal knowledge of product safely law; the absence of a label could either denote the absence of listed chemicals or the absence of a manufacturer who cares at all:
Of course, you may be in less danger with a clearly labeled item than you would be with unlabeled products, even if the overlabeling phenomenon leads to some false positives. Companies that are willing to comply with the law by warning buyers are probably providing safer goods than unknown manufacturers from third-party sellers overseas, since the latter may not be following federal guidelines for safety for things like lead paint or cosmetics additives.