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.
OK, I did it. The rice vinegar was definitely the biggest difference. A pinch of MSG (Accent) sealed the deal. I found a recipe on Serious Eats. I tasted after every addition, saving the MSG for last. It wouldn't be as close having skipped anything that I used.
I didn't have real hon-dashi or Japanese mustard, but I had some instant miso soup powder and some Colman's mustard. It was close enough. I also used just rice vinegar, I didn't have any malt vinegar.
I don't have any real Kewpie to do a side by side comparison, but I can tell this is pretty close. It's definitely much closer to Kewpie than Hellmann's.
Everything that has been said here was right.
Best Answer
This kind of eggless recipe depends on the lecithine in the soy milk to emulsify the oil with the water phase (the soymilk). Emulsification is a complicated process, and it does need all the starting conditions to be right.
From your description, it looks like your homemade soymilk either doesn't have enough lecithine, or it is in a form that isn't available for emulsifying the fat (maybe it is already bound to something else). This is actually to be expected, the process for making soymilk at home is relatively crude and you don't have much control over what happens to specific compounds in the soy. The food technologists at industrial producers can finetune their process to make sure their soymilk has a lot of lecithine available, and in the proper form to uphold a good emulsion. They are likely not doing it for mayonnaise purposes, but to prolong the "best by" date of their product (prevent their soymilk from separating in the fridge) but you are now seeing the side effects - not any old soymilk will do, you need one that is optimized for making emulsions.
As a side note, when producers list the ingredients on a processed product, they are not obliged to keep the ratio of derived products the same as in the natural ingredient - in fact, that would defeat the whole purpose (can you imagine a "soy milk" with the same ratio of solids to liquids as a soy bean?). So with the ingredient list you gave us, it is entirely possible that they used some soybeans to extract soy lecithine, purify it and do whatever is necessary for it to have its best emulsification powder, then extracted some soymilk from other beans and added the purified lecithine. The ingredients on the label can still say "soy", because the added lecithine is also "part" of the soy. So the most viable solution for your case might be adding extra soy lecithine to your mayonnaise. It will be tricky to hit the right ratio and best process, but it should give you a creamy emulsion.
In support of my line of argumentation, apparently not even all industrial soymilks are equally suited for this type of recipe, see this old question and OP's answer: Vegan mayonnaise never emulsifies?.