$\DeclareMathOperator\ker{ker}$Assume that $A$ is semisimple, so by Wedderburn's theorem, $A$ is isomorphic to 
$$M_{n_1}(k)\oplus M_{n_2}(k)\oplus\dots\oplus M_{n_r}(k)$$ 
where $n_1\leq n_2\leq\dots\leq n_r$ positive integers. 
Thus, two homomorphisms $f,g:A\to M_n(k)$ are equivalent if and only if $\ker{f}=\ker{g}$.

**Claim:** Given a norm $\|.\|_A$ on $A$, there exists $\varepsilon>0$ such that for any two homomorphisms $f,g:A\to M_n(k)$, $\ker{f}=\ker{g}$ whenever $\|f(a)-g(a)\|\leq \varepsilon\|a\|_A$ for all $a\in A$.

**Proof.** Let $p_i$ denote the identity of $M_{n_i}(k)$ so that $\{p_1,\dots,p_r\}$ is a set of pairwise orthogonal projections such that $p_1+\dots+p_r=1$. Let $$\varepsilon = \frac{1}{2}\min\Big\{\frac{1}{\|p_1\|_A},\dots,\frac{1}{\|p_r\|_A}\Big\}.$$

Now suppose $\ker{f}\neq \ker{g}$. Then, either $p_i\subseteq \ker{f}\backslash \ker{g}$ for some $i=1,\dots,r$, or $p_i\subseteq \ker{g}\backslash \ker{f}$ for some $i=1,\dots,r$. In either case, $\|f(p_i)-g(p_i)\|=1>\varepsilon\|p_i\|_A$.

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**Edit/Correction:** The original claim was incorrect. Rather than deleting the answer, we modified it to get some benefit, a partial answer out of what was written. Please see Yves de Cornulier's reply for a counterexample for non-semisimple $A$.