Perhaps most obviously, you just can't try a shoe on if you buy online. You can if you buy five pairs and return four, as some do, but it's a lot of hassle and a waste of resources for all the shipping back and forth.
Secondly, a significant portion of the population (tens of precent) have "problems" enough that most shoes in a store don't fit, meaning you have to try on a lot before you find one that does. And even if you don't have such issues, sizing between brands (and even styles) is so inconsistent that you usually have to try multiple different sizes to decide which one fits you. Having a 3D model paired with an AI system that tells you which shoes fit and what sizes to pick saves a lot of time for a lot of people, both shoppers and staff.
Finally, if you ever want to order custom made footwear, there is really no way around measuring your feet. And arguably the only reliably well-defined and repeatable way of doing that is to do a 3D scan to capture the shape.
My aunt would knit socks for me as a kid. I'd stand on a sheet of paper and my mom would draw the outline of my foot, and mail her the sheet. Worked great!
That method is actually used by many shoemakers when you order a pair of bespoke shoes (in combination with a series of tape measurements). It's not accurate enough to produce a fitting shoe in a single attempt, however, so what usually happens is they make a "test shoe" based the measurements and drawing, then modify the last after having you try that on. This procedure can be repeated more than once if needed.
Knitted socks are quite a lot more forgiving than leather shoes, fortunately, since the material accommodates by stretching when you put it on. I'm not surprised it worked well for you and your aunt!
Thanks this is interesting. Is part of the problem that people's feet change size with weight?
I have the impression that this is half the battle with finding shoes myself. Probably the bones are the same size left and right, but I think they deform differently as I stand (or walk) in a way that depends on the contours of the shoe... and that sounds like a problem that would be hard to model with laser scanning or whatever.
You're right that the feet deform significantly when you stand on them. For that reason you usually measure them load-bearing (i.e. you stand up with even weight distribution on your feet).
I think it's more likely that it's the deformation of the shoe that is uneven than that the shoe affects the deformation of the foot much. An exception would be if it has a built-up sole with support for the arches, preventing the arches from collapsing.
Either way, it's a complex process that is hard to model, just like you say. The way we're getting around it is by collecting various data on what people end up liking and then infer the properties of the shoe, rather than trying to explicitly model it. So the solution is a combination of 3D scanning and machine learning.
Also worth mentioning, by the way: most people actually have slightly different size and shape on their left vs. right foot. For about 50% of the population the length differs by more than half a US size. Being perfectly symmetrical is the rare case.
This may be mostly irrelevant though because unless you have problems with your feet you can tell if shoes fit simply by the feel.