Does multicollinearity among control variables matter? I am conducting a regression analysis between $X$ and $Y$, where $X$ is the main independent variable. However, I want to control for several variables that are related to $Y$. For example, my dependent variable is loan default and $X$ is a unique independent variable, but I also want to control for loan characteristics such as interest, maturity, etc. Note, I have only one independent variable. Does multicollinearity matter between control variables or is it only for independent variables?

Below, I will presume that we use ordinary least squares linear regression.
If you have multicollinearity only among the control variables, let's call them $Z_i, i=1, \ldots, k$, but the treatment variable $X$ is not part of the subspace spanned by the control variables, then you can use linear regression to obtain the effect of the treatment $X$ on the outcome.
Here is a figure which might help with the geometric intuition:

We consider the situation of a data set with only two data points, but this is easily generalizable to higher dimensions. Also, we presume that $k=2$, i.e. we have two control variables, $Z_1$ and $Z_2$. The vectors in the figure symbolize the columns of the design matrix, so the red vector $x$ is the column vector of the treatment $X$, the purple vector $y$ is the column of the outcome $Y$, and $z_1$ and $z_2$ are the columns of the covariates. As required, $z_1$ and $z_2$ are collinear and $x$ is not part of the subspace spanned by $z_1$ or $z_2$.
Clearly, to obtain a linear combination of the outcome $y$, we need to obtain the black cross by a linear combination $\lambda_1 z_1 + \lambda_2 z_2$ and then add a multiple of $x$ (the dotted red vector). For the linear combination $\lambda_1 z_1 + \lambda_2 z_2$ there are infinitely many possibilities. The linear regression algorithm will return one of them and, maybe, a collinearity warning. However, we are not interested in the coefficients of the covariates, but in that of the treatment $x$, and that one is clearly unique, as long as $x$ is not part of $\operatorname{span}(z_1)$.
If, on the other hand, $x$ would be collinear with $z_1$ and $z_2$, i.e. $x\in\operatorname{span}(z_1)$, the situation would be as follows:

Now, the best linear approximation of the outcome with the collinear vectors $x, z_1, z_2$ would be the orthogonal projection $Py$ of $y$ to $\operatorname{span}(z_1)$. Again, we have infinitely many solutions, but this, time, unfortunately, the coefficient of $x$ is not unique anymore, either. Thus, in this case, the real causal effect of $X$ on $Y$ cannot be identified from observational data and we would have to resort to interventions on the treatment.
