Is there any formal way to speak about fiber bundles using the language of exact sequences? After all, in some sense, they are similar:
$F\to E\overset{p}{\to} B$
The notion of an "exact category" (as a -not necessarily abelian- category equipped with triplets of objects forming "exact sequences") requires the category to be additive, which TOP is not.
This is too long for a comment; but it doesn't really give a formal way to talk about them as exact sequences, although it does give an analogy :
One possible way to make a connection is the following : a fibration sequence should be seen as an exact sequence.
A (Serre) fibration is a more general object than a fiber bundle, as every fiber bundle is a fibration, but not conversely.
Now given a group $G$ there is an associated space (or more precisely, homotopy type) called $K(G,1)$, which has no homotopy groups except $\pi_1(K(G,1),*) = G$ (up to isomorphism of course). It so happens that whenever you have a short exact sequence of groups $1\to H\to G\to Q\to 1$, you can realize it as a fibration sequence $K(H,1)\to K(G,1)\to K(Q,1)$ which, on $\pi_1$, induces the correct maps.
More generally, if you have a fibration sequence $F\to E\to B$, you have a long exact sequence of homotopy groups : $\dots \to \pi_{n+1}(B)\to \pi_n(F)\to \pi_n(E)\to \pi_n(B)\to \dots \to \pi_1(E)\to \pi_1(B)$ that extends a bit to the right as an exact sequence of pointed sets $\pi_1(B)\to \pi_0(F)\to \pi_0(E)\to \pi_0(B)$ (still more generally you have long exact sequences of pointed sets by taking homotopy classes of maps from any CW complex)
In this sense, fibrations can be seen as the homotopical analogue of (short) exact sequences; and fiber bundles can be thought of as particularly nice fibrations.