Refuse to read directly from his book, but from the extract above Peterson argues that giving rights to a person/animal has a direct impact on another person individually. Specifically in his argument for medical care, if one person is ill the responsibility to provide medical care would fall on one individual.

I'd argue that our rights result from the shared values of the societies that we participate in. The burden of those rights are shared by all members of the society. An example being: Large majority of European inhabitants value health and believe medical care should be accessible by all, European societies see health/medical care as a right, burden of providing healthcare falls on society as a whole.

Key difference is that no-one is anyone else's slave.

Even following his argument, what happens to children or other people who may not be able to "shoulder the burden" in his model? Does he lump them in with the animals who in his view do not have any rights?

So if society values animal welfare, they will have rights and the burden of giving those rights will fall on society.