Let me offer a "perturbation" argument making this plausible. Consider the equation
$$ y''(x) = 0. $$
The associated characteristic equation is $x^2 = 0$ and so it has a double root $x = 0$. Instead of solving the original equation, let us modify the equation slightly so that the problem doesn't occur and then see what happens. So instead of the original equation, consider the equation
$$ y''(x) - \varepsilon^2 y(x) = 0 $$
where we think of $\varepsilon$ as a small parameter. The associated characteristic equation is $x^2 - \varepsilon^2 = (x - \varepsilon)(x + \varepsilon)$ whose roots are $x_{1,2}(\varepsilon) = \pm \varepsilon$.
Hence, the modified equation has two linearly independent solutions
$$ y_{1,\varepsilon}(x) = e^{x_1(\varepsilon) x} = e^{\varepsilon x}, y_{2,\varepsilon}(x) = e^{x_2(\varepsilon) x} = e^{-\varepsilon x} $$
and the general solution has the form
$$ y_{\varepsilon}(x) = c_1(\varepsilon) y_{1,\varepsilon}(x) + c_2(\varepsilon)y_{2,\varepsilon}(x) = c_1(\varepsilon) e^{\varepsilon x}(x) + c_2(\varepsilon)e^{-\varepsilon x} $$
where $c_1(\varepsilon),c_2(\varepsilon)$ are arbitrary. As we take $\varepsilon \to 0$, the two different roots $x_{1,2}(\varepsilon)$ tend to the same root $x = 0$ and the two linearly independent solutions $y_{1,\varepsilon},y_{2,\varepsilon}$ tend (whatever that means) unfortunately to the same solution $y \equiv 1$ of the original equation.
How can we generate another solution? Instead of looking at the limit of $y_{1,\varepsilon},y_{2,\varepsilon}$ and get the same solution, we can look at the limit of $y_{\varepsilon}$ and try and choose the constants $c_1(\varepsilon),c_2(\varepsilon)$ so that we will get a solution other than $y \equiv 1$. An indeed, consider
$$ y_{\varepsilon}(x) = \frac{1}{2\varepsilon} e^{\varepsilon x} - \frac{1}{2\varepsilon} e^{-\varepsilon x} = \frac{1}{2\varepsilon}(1 + \varepsilon x + \frac{\varepsilon^2 x^2}{2!} + \dots) - \frac{1}{2\varepsilon}(1 - \varepsilon x + \frac{\varepsilon^2 x^2}{2!} + \dots) \\
= x + \frac{1}{2\varepsilon}(e^{\varepsilon x} - e^{-\varepsilon x}). $$
Here, as $\varepsilon \to 0$ we have $y_{\varepsilon}(x) \to x$ and this gives us another solution $y(x) = x$ for the original equation.
Warning: What I described above is not a rigorous argument and there are quite a few problems in making it rigorous in full generality but this is an example of a perturbative argument which is quite common in physics. We use the extra freedom in the choice of constants and try and set them up so that we other solutions. This explains how the freedom of multiplying by constant can generate extra solutions.
