Are there elements of sets that are no sets in Zermelo Fraenkel Set theory? I've seen the ZFC formalised in a lecture where the lecturer introduced, part by part, propositional logic, 1. order logic, and then zermelo-fraenkel-set theory. The lecturer didn't introduce any notion of identity ("=") in the part about 1. order logic, and defined identity in the part about set theory.
There, two sets where defined equal when from something being an element of the first set followed that it must be an element of the 2nd set as well, and vice versa (the definition captures what I read would be the "axiom of extensionality", later on).
However, in the further proceeding of the lecturer (here:  https://youtu.be/AAJB9l-HAZs?t=4456), the lecturer  used the "=" sign, and the notion of identity, not only for sets, but also for elements of sets.
Did he, in this moment, assume that every element is a set? And does this mean (for the further use of the ZFC) that I can only use ZFC to describe "collections" of entities that are as well a set?
 A: In ZFC, everything is a set, although we do use shorthands like $15$ and $G$ when we want to refer to mathematical objects that "we know" aren't sets. The idea is that all mathematical objects can be coded up as sets, so we forget that there are any other mathematical objects than sets.
For example $0$ can be implemented as $\emptyset$, $1$ can be implemented as $\{\emptyset\}$, and in general the number $n$ can be implemented as the set of all smaller numbers. (This is just one possible choice of implementation, but it turns out to be convenient.) Given an encoding of the natural numbers into set theory, we can then construct e.g. the reals using your favourite quotients of appropriate products. This allows ZFC to talk about real numbers (although it does so in an extremely unnatural way).
The general point is that by removing all the objects of maths other than sets, we get a simpler theory which we can talk more easily about.
For set theories which contain non-sets, look up "urelements" a.k.a. "atoms".
