Cossack uniforms were of different colors. Lisa’s link isn’t working so I don’t know which uniform Alexei is wearing, but I would guess it’s the one from the Emperor’s Own Cossack Escort, with a long red “Circassian”coat with a special Caucasian gold and silver braid (for officers), a white shirt and tall black Astrakhan wool cap. This was the dress uniform, worn with silver epaulettes: http://www.xenophongi.org/rusarmy/shenk/shenk6bs.htm
The ordinary dress was dark blue with a red shirt and shoulder boards instead.
Some Cossack cavalry regiments (the ones from the Caucasus) had a similar pattern uniform, with a long wool coat, either dark brown or black, with shirts of various colors. For example: http://www.xenophongi.org/rusarmy/shenk/shenk26bs.htm
There were other Cossack regiments though, from the Ural, the Don, the Steppes, Siberia, etc. which had a different uniform than the regiments from the Caucasus and wore a shorter buttonless tunic (various colors) with silver braid (for officers). The were 3 regiments of that type in the Guards: His Majesty’s Guard Cossacks (red tunic), H.I.H. the Heir Tsesarevich’s Own Atamansky Guard Cossacks (light blue tunic) and the Combined Guard Cossacks (formed with the Ural Cossack squadron - raspberry tunic - and detachments from some other Cossack regiments which had orange-yellow tunics). Ural Squadron officer seen here (with ordinary dark blue dress on the left) : http://www.xenophongi.org/rusarmy/shenk/shenk8as.htm
The Line regiments (not belonging to the Guards) had identical tunics, either dark blue or dark green, with various piping colors. Here is a Don Cossack regiment: http://www.xenophongi.org/rusarmy/shenk/shenk25as.htm
During the War, all Cossacks wore field uniforms of various shades of grey or khaki green instead of bright colors. Here’s an example of the uniform worn by regimenst from the Caucasus: http://www.xenophongi.org/rusarmy/shenk/shenk26gs.htm
Tsar Nicholas II and Alexis often wore a similar uniform at Stavka.
I have a few color prints of Alexis in Cossack uniforms I will try to post later.