well more than emotional compatibility there is compatibility period. emotional is only one aspect of it. There are value systems and beliefs and other things. such as how one expresses love to another and/or expects to be loved. There are different types of expression and this is where the most important aspects of compatibility are.

If one is a kinesthetic person that loves and feels love by touch but the other person does not respond to this type of expression it can be a problem, especially if the non kinesthetic person actually has an aversion to too much touch. One person could be trying to care for the other by touch, hugs, snuggles etc but the other person does not feel loved but may be put off by it, it results in both partners feeling unloved. The touchy person will feel like they are being rejected too. This is just one example.

the other types of expression that I can think of on the fly here are giving and servicing. where one would be the type that shows love by showering with gifts and the other is one that expresses love by doing things for their partner.

I know there are probably more types of love expression but that's all I can think of right now.