By Assertation 8.3 in Essay IV of "Foundational Essays on Topological Manifolds, Smoothings, and Triangulations" by Kirby and Siebenmann,
the following two definitions of a universal bundle $\gamma$ (over a space $BG$) are equivalent "for most types of bundles":

(*) For every bundle $\xi$ over a base space $B$, every closed subset $C \subset B$ and every morphism $\varphi:\xi_{|U} \to \gamma$, where $U$ is a neighbourhood of $C$, there is a morphism $\psi: \xi \to \gamma$ such that $\psi_{|C} = \varphi_{|C}$.

(**) For every bundle $\xi$ there is a classifying map $f:B \to BG$ such that $\xi \cong f^*\gamma$ and this $f$ is unique up to homotopy.

The first property implies that the classifying morphism $\xi \to \gamma$ is unique up to homotopy (take $B = X \times [0,1]$, $C=X \times \{0,1\}$ and $U=X \times [0,1/3) \cup (2/3,1]$), whereas the second property only requires that the base map of such a morphism is unique up to homotopy.

Often only (**) is proven (for example for vector bundles in Milnor-Stasheff "Characteristic Classes"), but by the statement all these (**)-universal bundles are as well (*)-universal.
This implies for instance that every automorphism of the universal bundle $\gamma \to \gamma$ is homotopic to the identity.

As it was mentioned in one of the comments above, I also want to regard the special case of the trivial bundle $\xi$ over a base $X$.
For simplicity, assume $X$ is the point $*$.
If we fix a base map $* \to BG$, then there are $[*,G]=\pi_1 G$ many morphisms over it, up to homotopy with this fixed base map.
(Here $G$ is the structure group of the type of bundles, which we regard. So $Gl_n$ for vector bundles.)

Note however that $\pi_0 G = \pi_1 BG$, so every element $g$ of $[*,G]$ corresponds to a loop $c:[0,1] \to BG$. If we now regard a morphism $\psi: \xi \times [0,1] \to \gamma$,
which has $c$ as its base map, then the morphisms $\psi_{|\xi \times \{0\}}$ and $\psi_{|\xi \times \{1\}}$ should differ by $g$, so
$$ \psi_{|\xi \times \{1\}} = \psi_{|\xi \times \{0\}} \cdot g.$$
This means that if we do not fix the base map of the classifying morphism $\xi \to \gamma$, there is really only one up to homotopy.

I interpret "for most types" as: We can take every kind of bundle (fibre bundle, microbundle etc.) but we should not regard bundles with an orientation or a framing. Or, at least, then we have to require that the morphisms preserve this extra structure.
Otherwise, the oriented line bundles would yield a counter-example.

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