The fact that the test for linearity suggests that the system is linear ***if*** we only consider real valued scale factors shows that the system would be linear if viewed as a ***real-valued*** multiple-input multiple-output (MIMO) system. We could model the system as one with two inputs and two outputs. In that case, the transfer function matrix is just a constant diagonal matrix:

$$\mathbf{H}=\left[\begin{array}{ll}1 & 0 \\ 0 & v\end{array}\right]\tag{1}$$

However, if viewed as a single-input single-output complex-valued system - as implied in the test question - , the system is not linear because the linearity test fails if we consider complex-valued scale factors. This is correctly pointed out in [Hilmar's answer](https://dsp.stackexchange.com/a/86618/4298).

I'd like to show another way to see that the given complex-valued system is not linear. First, note that the given system is cleary time-invariant. Consequently, if it were linear, it would need to be a linear time-invariant (LTI) system, and as such its output must be computable from its input via convolution with a (complex-valued) impulse response

$$h(t)=h_R(t)+jh_I(t)$$

With $x(t)=x_R(t)+jx_I(t)$ being the complex-valued input, the corresponding output is

$$y(t)=(x_R\star h_R)(t)-(x_I\star h_I)(t)+j\big[(x_R\star h_I)(t)+(x_I\star h_R)(t)\big]\tag{2}$$

From the specification of the system, we require the output to be

$$y(t)=x_R(t)+jvx_I(t)\tag{3}$$

Comparing $(2)$ with $(3)$ shows that there exists no impulse response $h(t)$ satisfying $(3)$. Hence, the system cannot be linear.

A complex-valued system can be viewed as a MIMO system. However, the transfer function matrix is very specific, and we can't choose the matrix entries independently. The transfer function matrix of a complex-valued system has the form

$$\mathbf{H}=\left[\begin{array}{ll}H_R(s) & -H_I(s) \\ H_I(s) & H_R(s)\end{array}\right]\tag{4}$$

where $H_R(s)$ and $H_I(s)$ are the transfer functions of the real and imaginary parts of the impulse response. Comparing $(4)$ with $(1)$ shows again that the given system cannot be implemented by a linear complex-valued system.