A code point is an integer value that refers to a specific character in Unicode. If two variants share the same code point, they are effectively the same from the computers point of view. If the code point is different, it means that the characters can be represented differently (but it depends on the font). More here if you’re interested!

The examples you brought up have different code points (otherwise they couldn’t be displayed as different characters without resorting to using pictures). And yes, this can be a bit frustrating, but you’ll get used to it! If you look at your first example, you can see that they are actually made up of the same components, but stacked vertically in one case. There are in fact a whole series of such variants.

I’m not sure what you mean with your follow-up question about traditional characters? In your specific case, one is standard in Hong Kong, the other in Taiwan, but they have probably coexisted for a long time, different government institutions just chose different variants for their respective standards, for some reason!

And how these different variants survived coexisting as traditional forms?

The most frustrating part is that sometimes the difference between variants is bigger than the difference between different words 😅

example
裏 & 裡 – variants of the same traditional character
力 & 为 – totally different words, both simplified characters

Which characters in particular do you refer to? The ones on the right use the standard Kaishu fonts issued by the Ministry of Education (Taiwan). This may or may not reflect the way your wife (or any other individual) actually writes.