1

Here it goes in context: "if you agree to treat like cases alike and that a difference in treatment requires a morally relevent difference, then you have to identify the differences that justify treating non-human animals in ways that we would never subject humans to."

3

The word like is an adjective here, meaning "similar":

If you agree to treat similar cases similarly and that a difference in treatment requires a morally relevent difference, then you have to identify the differences that justify treating non-human animals in ways that we would never subject humans to.

This phrase, "treat like cases alike", is a very old phrase. Judging by a quick Google search, it dates back to Ancient Greeks:

Aristotle (384–322 B.C.E.) argued that equality requires that equals be treated equally: we must “treat like cases as like.” (Google Books)

From the Equality article in the Stanford Encyclopedia of Philosophy:

When two persons have equal status in at least one normatively relevant respect, they must be treated equally with regard to this respect.This is the generally accepted formal equality principle that Aristotle formulated in reference to Plato: “treat like cases as like” (Aristotle, Nicomachean Ethics, V.3. 1131a10-b15; Politics, III.9.1280 a8-15, III. 12. 1282b18-23). Of course the crucial question is which respects are normatively relevant and which are not.

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