Now, I have not counted lately, but I've got about 100 or more different sets of attachment feet - over half are of the top clamp variety - and most are unique in some way.
There are two basic widths of the gap - 1/8" (I've measured .120 to .127)and 3/16" (I've measured .179 to .182). Free, and the machines they made, are the only ones I've found that take the narrow gap. All the rest take the wide gap. ** if you have a New Home that takes a narrow gapped foot, it was made after Free took them over - and about the only thing "New Home" about it is the name.
There are visually different - you can easily tell from a picture which one you are looking at. Pay attention to the ratio of metal on each side of the gap compared to the gap itself.
The next basic thing is distance from the front edge of the gap to the needle hole. If the hole is Centered, there are going to be two basic distances, 1/2" and 5/8" White is the only machine that take the larger distance.
There is one other difference, though it does not come into play that often - overall height of the foot.
Older Nationals are normally shorter than those for newer national machines, but I've not found it's a huge issue.
However, the far left is a foot for Standard. And while a National foot will fit and line up, when you lower the foot, it does not actually come in contact with the feed dogs.
The thickness of the metal of the foot is not important to the fit - the narrow hemmer foot and basic foot that would come with the machine often much thicker metal than the accessory set.
When you start adding in the earlier machines that didn't stay in business that long after 1900 (Demorest, Household, Domestic (before White got them), Florence, Davis) and some others, there are some other differences, most notably an offset of the needle hole to the left or right of the presser bar.