Η σούπα έχει κρεμμύδι.

Indeed. Κρεμμύδι is a word of Greek origin, which is why it doesn't remind you the word for onion in another language. I can't think of a way for someone to be able to remember this word. Some learners tend to memorize some words by memorizing a part of the word that sounds kind of familiar to them, so by memorizing a funny phrase containing that word.

For example, ντυμένος σαν κρεμμύδι - Dressed like an onion, is a phrase that a lot of people use in the winter, to describe someone wearing many layers of clothing, just like the many layers an onion has. but I'm not sure if that's helpful for everyone :P

For me this was helpful: "creamy the (κρεμμίδι) onion". Though an onion usually is not creamy (unless you smash it and whip it all up, then it becomes creamy consistency), so this made a funny connection to me :)

You didn't miss a thing ;D

Έχω directly translates to have, while περιέχω directly translates to contain.

In Greek, when we talk about food and its contents, it's pretty common to use the verb έχει, in sentences like "έχει αλάτι/κρεμμύδι/κρέας" etc. Περιέχει is a bit less commonly used, at least for food. In English though, the most common verb to use would be contains instead of has. So because it's an awkward kind of sentence, we decided to include both alternatives. (At least in Greek. We will probably remove the english alternatives with has, just because they sound weird)^.^