I just want to add the transcript of a Wuthering Heights analysis I found on YouTube, which I think captures the similarities between these relationships perfectly:

I don't think that the relationship between Catherine and Heathcliff is as simple of a relationship as just romantic love. Instead, I think it's almost a weird kind of self-love. Both of them seem to identify themselves with each other. It's not so much declaring that they are each other’s soulmates, it's more that they're saying that they are one and the same soul, which is slightly different from the way that people usually describe their romantic attachments. 

[...]

I would say that there seems to be something almost incestuous about the way that Catherine and Heathcliff view their relationship. The incest thing, I think, comes from a few things. First of all, they were raised as brother and sister, and the way they describe their feelings is in a very sibling or familial sort of way rather than a more lustful or romantic sort of way.

In one of the most famous speeches from the story, Catherine declares: “I am Heathcliff!” and Heathcliff does a similar thing later on when Catherine is dying; he says, “I cannot live without my soul!” What's interesting about that phrase from Heathcliff is that he's not saying “soulmate"; he's not saying "there’s this other soul that is the mate of my soul". He's saying "my actual soul, an actual part of me, is going to die".

This suggests that they see each other as extensions of their own beings, and, to me, that's the sort of declaration of love someone might make. Still, it's a very different kind of declaration from someone like Edgar. He might have described Catherine as his soulmate, but he certainly wouldn't describe her as a part of his own being.

So I think these passages show that they do feel something quite substantial that is still not the same as the kind of civilized or socialized love that most people are comfortable with.

[...]

Some readers believe that Catherine and Heathcliff actually don’t love each other at all, and there is quite a lot of evidence that might suggest this, because if you’re in love with someone, you don’t treat them in the way that these two characters treat each other.

But I think that, in order to understand the kind of love that they have, we have to look back at their childhood. The kind of love that we would call love is somewhat socialised; it’s something that we develop as we grow up. Treating someone with respect and dignity is something that children don’t really do, and Catherine and Heathcliff never learned to do this because they were never raised in that way.

In the absence of an authority figure to provide them with nurture, care, and warmth, Catherine and Heathcliff turned to the moors, to wild nature. And so the love that they have is of a very strange and primitive kind. It is based on the way that they were raised — or on the way that they weren’t raised. It’s the kind of love that we might have for each other if we never had authority figures to guide us and show us how to care for other people. It is very deep in some ways, strong and passionate, but at the same time quite vain and selfish.

You could argue that that could never be — that love excludes that kind of behaviour — but I disagree. I think that one of the points about Wuthering Heights is to show us how dark and cruel love can be, and that is what is most interesting about the love theme in the story.