I do not know if there is a name for this problem. However, we can view this in terms of the usual operator norm.
Given $A \in \mathrm{Mat}_{k \times n}(\mathbb{R}), B \in \mathrm{Mat}_{k \times m}(\mathbb{R})$, consider for $x \in \mathbb{R}^n, y \in \mathbb{R}^m$:
\begin{align*} \underset{\substack{ \lvert \lvert x \rvert \rvert = 1 \\ \lvert \lvert y \rvert \rvert = 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \lvert \lvert Ax + By \rvert \rvert \\ &= \underset{\substack{ \lvert \lvert x \rvert \rvert \leq 1 \\ \lvert \lvert y \rvert \rvert \leq 1}}{\mathrm{sup}} \, \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert \\ &= \underset{\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty = 1}{\mathrm{sup}} \left \lvert \left \lvert \begin{pmatrix}A & B\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} \right \rvert \right\rvert, \end{align*} where $\left \lvert \left \lvert \begin{pmatrix} x \\ y \end{pmatrix} \right \rvert \right \rvert_\infty := \max(\lvert \lvert x \rvert \rvert, \lvert \lvert y \rvert \rvert)$ is the supremum norm from viewing $\mathbb{R}^{n+m}$ as the product $\mathbb{R}^n \times \mathbb{R}^m$. Observe then that the last expression is $\lvert \lvert (\begin{smallmatrix}A & B\end{smallmatrix}) \rvert \rvert_{\mathrm{op}}$, where $\lvert \lvert \cdot \rvert \rvert_{\mathrm{op}}$ is the operator norm induced from $\lvert \lvert \cdot \rvert \rvert_\infty$ on $\mathbb{R}^{n+m}$ and $\lvert \lvert \cdot \rvert \rvert$ on $\mathbb{R}^k$.
