Maybe I'm reading too much into this, but the Cyrillic Ш/ш (sha) looks like the Chinese 山 (shān) and have similarity in pronunciation. Is there any actual relation between the two?
No. The Cyrillic script is based on the Greek script, and some other local scripts like Hebrew. The basis for sha is thought to be the Hebrew letter
It's unlikely that shin is based on the Chinese character, either. It seems taken from the Phoenician alphabet, where the corresponding letter looks like a Latin
It's worth keeping in mind that at the time Cyrillic characters were invented, Russia did not yet control Siberia. So it was very far from China, and contact between China and the west was limited.
But maybe the Chinese character was influenced by these scripts?
It's not impossible, but the earliest versions of the Character look more like a triangle, so the simpler explanation is that it's just a picture of a mountain:
Considering the evidence, it seems unlikely that these characters are related.