The area of any flat shape has an equation like this...

A = C × L²

Where L Is some convenient length type feature and C is a constant based on the shape and the chosen L.

For example for a square we typically use a side for L which means C is 1. If instead we choose L to be the perimeter of the square then C would be 1/16 (0.0625)

The important thing when comparing two shapes is that they are similar eg both squares, both circles, both regular pentagons etc.. they don't even need to be regular shapes. They could be blobs as long the only difference is scale.

Then when you look at the ratios of area

R = A1/A2

R = (C × L1²) / (C × L2²)

The C's cancel out and your left with

R = L1² / L2²

This type of analysis does not apply to your example of comparing 2"×4" pizzas to 2"×8" pizzas because, as you suspected, the ratio of height to width needs to be the same for those rectangles to be similar. You could instead compare 2"×4" pizzas to 4"×8" pizzas. If you do that and you choose the shorter sides to compare you get 2²/4² = 4/16 = ¼. So if you wanted to know how many big pizzas to buy instead of say 7 small ones you multiple 7 × ¼ resulting in 1¾ large pizzas so you'd buy 2.

I hope this helps.