Yes since mayonnaise contains eggs the mayonnaise makes a great substitute for eggs in recipes that call for eggs.
For example when meatloaf calls for eggs you can use mayonnaise in the meatloaf instead of eggs.
But for cake or other pastry dishes I would not use mayonnaise in it as it may not taste good to you.
I tried using mayonnaise in a cake once and although it held it together the mayonnaise in the cake didn't taste too good.
So for a cake I would use eggs instead of mayonnaise but for baked goods that contain meat the mayonnaise actually makes the meatloaf and other baked meat taste good.
I use mayonnaise all the time when making meatloaf instead of just eggs and it tastes really good.