Egg mayo is the egg sandwich stripped to its shortened name, and the name tells you what it is: not a tea-tray construction, not a pairing, just the default. This is the sandwich every other egg sandwich on the shelf is a variation of, the one ordered without thought at a counter or pulled off a chiller shelf, where the colloquial contraction is itself the signal that nothing has been added and nothing needs explaining. Chopped hard-cooked egg, mayonnaise, salt, pepper, soft bread. The defining quality is that it has no defining feature beyond being correctly made, which makes it the hardest of the family to fake, because there is no herb, no leaf, no sauce to carry a weak version.
The craft is the bind and only the bind. Egg is chopped and held with just enough mayonnaise to make it cohere without turning to a wet paste: too little and it spills out the sides, too much and it slumps into a slick, and the entire sandwich is decided by getting that ratio right. The texture is a choice, finely mashed for a smooth, sandwich-shop filling or left coarser for distinct pieces, and either is legitimate as long as it is deliberate. Salt and white pepper go through the egg in the bowl rather than onto the bread, because seasoning the filling and seasoning the slice are not the same thing. The bread is soft and plain, buttered to seal the crumb against a filling that is by design slightly wet, pressed and cut clean.
The variations are the entire bound-egg shelf, every one of which is this sandwich with one thing added or one thing swapped. Egg and cress adds a peppery shoot; egg and chive an allium line; egg and salad cream changes the binder for a sharper one; egg and tomato adds acidity and a water problem. The full-name egg mayonnaise is the same filling in a more formal register. Each deserves its own article rather than being crowded in here.