The reason humanity won't starve, in the end (as some others suggested) is that humanity is very, very good at surviving, and is absolutely capable of being vicious, brutal, and cruel when necessary.
They will utilize your loophole. The major, gaping loophole that has been overlooked. There is no need to treat animals well, to give them an actual high standard of living - they simply must be treated like humans and given a similar/equivalent standing. Animal rights are, in your world, equivalent to human rights - but human rights are not set in stone! Human rights already vary by a lot of factors, including government, area, physical environment, historical period... I expect to see minimizing or outright discarding human rights before I see people dying en masse to accommodate animal rights. If we can't change the legislation about animals being treated equally, we will slide everyone's standard of living down until the levels are "survivable" - and practice "some are more equal than others" whenever and wherever we can.
So, fairly soon after implementation - there will be changes that legalize human slavery, cannibalism, and murder - all of which legalize the exploitation and consumption and slaughter of animals. The definitions of "harm", and "adversely affecting health" (mental or physical) will be redefined to not cover any harm or damage to health we can't afford do without, like starvation of those wild (or "free"), or exploitation of the enslaved (in return for "care"). Probably it would start with making humans do the work because someone has to, then progress to the idea that if animals are protected form the things we protect humans from, they can be made to do anything we can make humans do. Then the dropping of human protections, allows the continuation of farming and meat processing, pharmaceutical testing (by removing all consent requirements, instead of excluding animals because they cannot consent), and a host of other preexisting systems that people will die without.
I expect in some areas this would be initially considered almost a formality - legalizing, say, cannibalism would be about letting animal meat consumption continue, or using slavery for nonpayment of taxes to continue with animal labor, and expecting that social taboo would keep eating or exploiting humans from appearing in practice... until, of course, somebody actually did (complying with whatever regulations are in place, of course). At which point, social anarchy in all its glory will commence, for a while. Anyone accustomed to a higher standard of living will make a point of targeting anyone who thinks the new law(s) are a good idea in any sense of the word - drastically dropping the number of supporters in a very short period of time.
In other areas... well, I said legalization, but it needn't be quite so formal - if not legalization, per se, I expect at least the removal of whatever systems currently make it criminal to do such things. If someone kills an animal (in self-defense or other directly survival-related reasons), and is tried for murder - next round has that judge/jury killed in "self defense" because the penalty is the same, and the danger is the same, right? If someone is starving, and the penalty for eating the chicken is the same as eating a human, than the law enforcement who would have enforced the penalty for the first will end up in someone's pantry.
Survival can't be legislated away so easily.