That doesn't imply any sort of objective morality. It implies there are multiple possibly workable moralities that conform roughly to the general popular consensus of what is right. There are also a infinite number of possible moral systems that don't do so.
For example you can easily construct a moral system where child rape is perfectly acceptable but not without deviating strongly from the norms we abide by to the point that invoking "morality" in such a case is virtually useless.
If I issued the same challenge to provide a coherent case that child rape is wrong, it would be easy to do so from multiple angles. You could easily make utilitarian or consequentialist arguments or deontological arguments along those lines.
I only add this constraint that the moral system you will have to construct to justify meat eating will necessarily be less compelling to most of us and more at odds with the values we generally hold because without this constraint it's pointless to talk about morality at all and you can literally justify any and everything.
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