What does the notation $L/K$ for a field extension exactly specify? In group theory, such an object would be the group of cosets of a normal subgroup $K$ of a group $L$ and a similar usage exists for quotient rings.
In field theory however, I have the impression that the object $L/K$ is the field $L$ itself and that the notation only tells us that we want to talk about elements of the subfield $K$ as well. Is this correct or did I miss something?
On
One initial way to view the notation $L/K$ is not as an object but as a description or a statement, which says that the field $L$ is one which contains $K$. So your impression is correct. If you want to view it as an object, you can think of it as a pair $(L,K)$ with $K\subseteq L$, or else as as a pair of abstract fields with an implicit monomorphism $K\hookrightarrow L$. Since we talk about features that extensions can have (being rupture or splitting fields, being normal or Galois, separable or inseparable, algebraic or transcendental, finite or infinite, etc.) it also makes sense to think of it as an object in its own right when one has enough experience to have perspective on all of these features.
A field extension $L/K$ is not a quotient in any sense. It just (confusingly) uses the same notation.
When we talk about a field extension $L/K$, we mean that $K\subset L$ as fields. Sometimes, you might find the terminology $L/K$ used to mean that there is a field homomorphism $$ K\hookrightarrow L $$
(exercise - why is this the same sort of thing?) But it is never a quotient.
We like to talk about $L$ being a field extension over $K$, so we use the normal notation for over - $/$ (as in $2/4 = 2 \textit{ over } 4$).
At least at a basic level, writing
tells us nothing more than writing
The distinction is that the extension $L/K$ has interesting structure in its own right (for example, $L$ is a $K$-vector space). You'll get used to this as you do more field theory.