I found the congruence in the title today (where $\tau(n)$ is the Ramanujan tau function), and I wonder if this is new or already known (or easy consequence of known results). I read the wikipedia and found two almost same looking congruences, just with different modulus:
$$ \tau(n) \equiv n \sigma_9(n) \,(\mathrm{mod}\,7)\quad \text{for }n\equiv0,1,2,4\,(\mathrm{mod}\,7) \\ \tau(n) \equiv n \sigma_9(n) \,(\mathrm{mod}\,7^2)\quad \text{for }n\equiv3,5,6\,(\mathrm{mod}\,7) $$
So the main difference is that mine includes $2,3,5^2$ in the modulus, and the second one above has $7^2$ instead of $7$.
It is known. See (12.6) on page 24 of https://www.mat.univie.ac.at/~slc/wpapers/s42berndt.pdf