CacherChick is an expert, so anything I say defers to her and builds on it:
(A) That left pipe
should not be where the water comes down from above, but it
might be. It shouldn't be, because (1) that washer standpipe needs a vent, so it should be a vent, (2) the way the pipe is attached is not via a fitting that would direct the flow of water from above, which would be required by code if it was going to carry drainage, (3) where that pipe is located is where a vent pipe should be and it is attached like a vent pipe, and (4) code would require that that pipe be used only as a vent, because "wet venting" (meaning a pipe that is used for sink runoff and similar limited drainage flow but otherwise serves as a vent) is not allowed by NY code for drainage that is flowing from one floor to a floor below (NY Plumbing Code Section 909).
This doesn't mean that during a remodel some enterprising non-plumber handyman (or a hack plumber or the homeowner) didn't attach the sink and shower drain to that pipe, and if they did, the problem that you are having is the predictable result.
(B)
If drainage
is going into that pipe, you probably have a clog at the spot to which you have pointed with the arrow.
(C) That washer standpipe seems to be too short. The correct height is between 18" and 30" from where it connects to the trap, which must be between 6 and 18 inches from the floor. In other words, the top of the standpipe has to be at least 24" from the finished floor, and as much as a foot higher than that (and higher if the trap is higher).
(D) It seems odd that if the two types of wastewater (shower/sink and toilet) are both going into the bigger pipe that you would get a backup only when you use the shower and not when you flush, although if you have a newer 1.6 gallon per flush toilet, you may be putting less water into that drain from the toilet than if you take a shower and thus it doesn't come out of the standpipe.
(E) As CacherChick says, the most likely problem is that the main drain to the sewer is backing up (or restricted) at some place below the point that that whole system connects together in your photos, and you should have someone come and snake (rod) out the main line, or water jet it. I don't know where in NY you are, but if you are on Long Island (Nassau/Suffolk), a respected plumber on here recommended Dan at Ranger Sewer to me (
www.rangersewer.com), and we used him several times at our family home in Nassau. He did a great job at a reasonable price for the quality of service. Among other things, we had a driveway drain that a couple of drain companies and a trusted contractor told us would have to be dug up and the piping replaced, because the long pipe from the drain to the cistern it emptied into was just too clogged with dirt and gravel, after efforts were made to rod it out. Dan came with his powerful water jet setup, and blew the thing out, with a lot of effort, in about half an hour. That was several years ago, and it still works like a champ. And his price for that was excellent, for what it was. Other people we referred him to have been more than satisfied. He is very meticulous, and takes each job very seriously.
PS If you are interested in learning more about your home's plumbing, here are two handouts for homeowners (one with illustrations and the other with photos) that explain how things are supposed to go together. They are not based on NY Code, but most of what they say is the same as is required here.
Plumbing-and-Trenching-for-Homeowners---2018 (klickitatcounty.org)
Helpful Plumbing Hints for Residential Construction by Bert Polk Plumbing Inspector Lincoln County
