More than possible, and was my thought earlier, or moved. Other thought now is possibly shower added. But just guessing from the way drains are laid. Or just some plumbing issues and this was how it was fixed.
Either case what Im seeing is a tee (not directional sweep) with a valve above/before tank. Whether water flows A or B
(A) #2 is to tank and #1 is from shower, or
(B) #2 is from shower and #1 goes to tank,
nothing directs flow thru tee- water dumping down #3 will likely go into both legs. Unless the shower is elevated alot you would see water from sink. Hence I think A and valve was to keep water from coming up shower. Drain restricted more so. Especially, due to lay out drains are probably indeed full of crud, restricting flow.
But if valve does close line to/from shower there should be no water coming up?
Easy enough if you had a place to do it, gray tank empty (gray) dump valve open, run water thru sink, with inside valve closed (handle pushed in against body) water should freely run out. Open inside valve-(handle fully pulled out bout 3" as Wayne described) look to see if water coming up shower. If no water flows outside with inside valve closed and outside valve open, but does when you open inside valve-well thats a problem, cant imagine that. Or you could add water to shower with valve closed, water would back up pretty quick if valve closes shower line. It could be valve was added because theres no P trap in shower.
I could think of a half dozen plausible scenarios how this is plumbed. You have rubber seals on valves, generally brass sensors in the tank(s) draino can be a problem, snake anywhere near a valve can ruin it, if it ends up in tank, well.
My fix would be start with water running both sink and shower alternating plunger between, if vents caps removable garden hose down it.