Protein in mayonnaise…none

Starting from the basics: Mayonnaise, as you know, is a combination of water-based liquids, water-soluble ingredients, and lipids (fats/oils). Since water and lipids are immiscible, that makes mayonnaise an emulsion.

Because the droplets (of fat) suspended in an emulsion are not actually dissolved, the properties of that emulsion depend entirely on the size of those droplets and their dispersion. The most likely reason that your mayonnaise tasted like oil is that it actually was pure oil in spots.

The technical term for this is flocculation.

(source: Cube Cola)

This is probably what happened to you - it's possible that if you had really poor dispersion, you might have even been closer to the "coalescence" stage.

To use a more tangible example, consider what happens when you dissolve flour or corn starch in cold water, then heat it. The starch gelatinizes and you end up with a fairly uniform, thick paste. Now think of what happens if you toss it into hot water; you'll tend to end up with something that isn't uniform, instead you'll end up with big globs of cooked flour floating around in thin, cloudy water.

Keep in mind that the chemistry is completely different with an emulsion - in fact, there technically is no chemistry happening with an emulsion until emulsifiers come into the picture - but the concept is the same. You might not be able to see those globs of oil floating around in the water as well as you can see the globs of flour, but if you didn't get proper dispersion and suspension, they're there, and they will taste exactly how you'd expect a glob of pure oil to taste.

Traditional mayonnaise uses raw egg yolk (containing lecithin) and mustard (containing mucilage), both of which act as emulsifiers. These are called "emulsifiers" mainly because they help the emulsion to stay stable, which is why the store-bought mayonnaise doesn't separate (it also probably has a few extra additives). However, they aren't all that helpful for getting that initial dispersion; the most efficient way to do that is to let small drops of oil into a liquid that is being constantly and uniformly agitated.

You can do this by stirring, but an even better way is to use an immersion blender with emulsifying blade. Note that this is not the flat aerating blade that is often confused with the emulsifying blade, nor is it the star-shaped liquefying blade that is the default on most sticks and many manufacturers confusingly call an "emulsifying" blade. The one you want looks a bit like a hubcap; it's flat with several slits or holes and is sometimes also called a "smoothie blade" or "whisk blade":

or

(the one I'm talking about is the bottom left)

These things are perfect for preparations like mayonnaise, but if you don't have one, you can get halfway decent results with a wire whisk. You'll just need to use a lot of elbow grease.

If you get really good dispersion, and use sufficient emulsifiers such that the emulsion doesn't separate too fast, then I promise you, your mayo won't have that "fatty" taste and it will be 1000 times better than the store-bought goop.

What about simply baking meringues? They're basically dried egg white with mostly consists of protein (Egg white nutrition data). You can simply take with a few as long as you can manage to keep them dry.