Suppose that $\rho$ and $\sigma$ are degree $n$ irreducible representations of a group $G$ over $\mathbb{C}$ and that for every $g\in G,$ there is a matrix $T_g\in GL(n,\mathbb{C})$ depending on $g$ so that $\sigma(g) = T_g^{-1} \rho(g) T_g$. Does there exist $T\in GL(n,\mathbb{C})$ so that $\sigma(g) = T^{-1} \rho(g)T$ for all $g\in G$?
I think the answer is no, but I'm not sure how to come up with a counterexample. The question basically asks if $\sigma$ and $\rho$ are necessarily equivalent representations. As a first step, it seems reasonable to find a specific pair of degree n irreducible representations satisfying the constraints in the question, but I'm not sure how to do so. One irreducible representation that could be worth considering is the sign representation of $S_n$.
If $G$ is finite, then an answer has been given in the above comment of @Just a user.
In fact, a similar proof works if $G$ is infinite: From the answers to this MathOverflow question it follows that $\sigma$ and $\rho$ are isomorphic.
To wit, if $I$ denotes the annihilator ideal of $\sigma\oplus \rho$, then $R=\mathbb{C}[G]/I$ is finite dimensional. Moreover, $\sigma$ and $\rho$ have the same character (over $R$). Hence, it follows from Theorem 7.19 in Lam's “A First Course in Noncommutative Rings” that $\sigma$ and $\rho$ are isomorphic.