The sidewall bulge looks bigger on a narrower wheel because the wheel is getting narrower faster than the tire is, and the only reference you have when you "eyeball" the bulge is how far out past the wheel flange it goes. You don't see that the rim is narrower. If you want to get really picky, tire diameter does increase minutely on narrower than measuring-width wheels (by something like 5% of the difference between actual wheel width and measuring width for the tire size).
I doubt that contact patch width varies enough to matter, particularly if you find yourself adjusting inflation pressure downward to minimize center-tread wear. In any event, the actual tire contact pressure against the pavement is far from being uniform (like any of the weight divided by inflation pressure calculations assumes is the case). The very edges of the tread are going to be lightly loaded (because the shape of tire shoulders curves upward away from the pavement) even when the outermost rib is generally more heavily loaded than the average over the entire contact patch. Unit pressures against the pavement within the contact patch can easily vary from some minimum number to 50% higher. It might look something like this ↓↓↓
Wheel width is defined between the bead seats, which are inside the flanges of the wheel. Typically, wheel flanges run about half an inch, so (as a made-up example) a tire that is claimed to be 11.2" wide when mounted on a 10" wide wheel would only have 0.1" of apparent bulge (what you see) because the outside of the wheel is ~11" wide. But relative to the 10" wheel's bead seat width, it's still 0.6"/side.
You can see that on the narrower wheel, the tire is narrower. But its sidewall bulges out more past its rim and i
s more sharply curved. This is my own spreadsheet which uses an approximate curve-fit for the sidewall shape. It's not perfect, but it is more useful than any of the online-available plots that ignore section width variation with wheel width. 255/45-18 on 8" (min-recommended) and 9.5" (max-recommended) wheels, if the text in the picture is too small or too obtuse.
Norm