Publons - is it a serious thing?
Here is their current web page, which in contrast to the Google+ page you linked to appears to be up-to-date.
Although this is the first time I've seen this site, my impression is that it's ethically okay and not a hoax. That said, it also seems quite unnecessary to me, and I would not bother to use it.
What will it yield scientists in the long run?
-- I would estimate:
With about 2% probability, the website will catch on and become popular. Probably this would be at the insistence of university administrators, who are often looking for new ways to measure and evaluate the performance of faculty.
The website will yield another hoop for researchers to jump through, and possibly (as the website claims) to increased recognition of researchers who do an unusually good job of peer review.
With about 98% probability, the website will largely be ignored, and the net effect on the scientific community will be nil.
In general, I would advocate ignoring advice originating from commercial publishers; personally, I would only consider using this site if I were urged to do so by scientists in my field or by administrators at my university.
It is probably legit, as it is owned by the behemoth company Clarivate, which does data analytics on all things science, engineering, and biotech. While it does not seem "unethical", it does seem like a business strategy to get peer review data and do analytics on it for profit.
I added one of my recent reviews, using the publisher's link, to test the site. Unfortunately, it shows the name of the paper that I reviewed in my profile. Although I could delete it, seems like you have to be very mindful about what you post and how you manage it, which equals adding more work to an already overworked schedule.
Peer review is a community service that we perform for the betterment of research. While it is one of the criteria used for academic promotion, it is not the most important, and therefore I do not think is worth the effort of using the site.
The other answers already give several interesting angles; I want to add one here. Publons simply counts the number of peer reviews, not their quality. Hence it promotes pumping this number at the expense of their quality. This is a direction that has become increasingly more popular in evaluating researchers, but if you think about it it's the exact opposite of where we'd probably want academia to go.
Calling it "unethical" is probably too much, but it's definitely not a line of thought that I wish to support.