Washing machine overflow
There's probably 50 years of lint build-up in the drain line that services this leg, causing it to be somewhat restricted. Draining a washer into a laundry sink buffers the high flow, which both makes that less of a practical problem (so long as it drains eventually) and also encourages the problem (since the water flows more slowly) - if it's backing up now, then the line beyond is probably in need of cleaning.
The vent line has some serious code issues by the codes I work under - apart from the question of how long your wet-vent is (on a trap that's absolutely testing the limits of venting to not get suctioned dry - washing machines are difficult) it is clearly going horizontal low enough that I doubt it's 6&quot; above the flood rim of the highest fixture on the floor served. Indeed, it would appear to be below the level of the standpipe, though that does involve interpreting between pictures, but I also doubt it's 6&quot; higher than the rim of the laundry sink I can't see.