Comparable ingredients in butter and mayonaise
Normally, there are no ingredients shared between butter and mayonnaise. The only ingredient in butter is milk, either fresh or cultured. The ingredients of mayonnaise are egg yolks, vegetable oil, mustard, water and acid. As you can see, there is nothing in common between the two. 

There are two exceptions when they may share something. 

First, you can get salted butter, and many mayonnaise recipes also incorporate some salt. Then the salt is the shared ingredient. 

Second, fats are interchangeable in the making of emulsified sauces, and home cooks will frequently call their sauce "mayonnaise" when made with butter instead of oil. Technically, this makes it a hollandaise derivative and not a mayonnaise, but many people won't know the difference. If you happen to eat "mayonnaise" made with butter, then the shared ingredient is the butter itself. With commercially bought mayonnaise, you can be practically sure that there is no butter in it.