In the context of the "divisor problem for arithmetical progressions",
there is the following article by Fouvry, Iwaniec and Katz, on "The divisor function over arithmetic progressions": http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6134.pdf.
The authors use Fourier series techniques
and Weil’s estimate for Kloosterman sums to obtain
$$
S(x,m,k)-D(x,m)=\sum_{n\le x, n\equiv k(m)}d(n)-\frac{1}{\phi(m)}\sum_{n\le x,(n,m)=1}d(n)=O((m^{1/2}+x^{1/3})x^{\epsilon}).
$$
We have the estimate
$$
\phi(m)D(x,m)=\sum_{n\le x,(n,m)=1}d(n)=xP(\log x)+O(x^{1/2}m^{\epsilon}),
$$
with $P(\log x)=\frac{\phi(m)^2}{m^2}(\log x+\gamma-1)+2\frac{\phi(m)}{m}\sum_{d\mid m}\frac{\mu(d)\log d}{d}$.

Werner Georg Nowak proved in $1984$ that for any given natural numbers $m$ and $k$,
$$
S(x,m,k)=\alpha x(\log x+2\gamma -1)+\beta x+ O(x^{35/108+\epsilon}),
$$
for constants $\alpha$ and $\beta$ depending only on $m$ and $k$.