"It is like with my cousin's dog."
I'm not sure the English sentence makes sense. Normally, we'd hear "It is like that with my cousin's dog"
Context would tell, but I think you could imagine one that fits in French at least.
One would never say this sentence in English. It would be 'It is like that with' or 'It's the same with....'
Given without context this is nonsense... this it shouldn't be part of the context-light component of this website.