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?.