Bad smell on clothes caused by plumbing?

I've never seen something like that before.

The most accepted form of plumbing requires a P-trap to prevent sewer gases from coming back into the house. In your case they might be flowing into your washing machine.

I haven't seen many washing machine to under sink connections before and the more typical thing is a long standpipe for a washing machine to pump the water into. I'd be fearful that the volume coming out of the washing machine would be too much for the sink assembly to handle in a short period of time possibly resulting in the water backing up into the sink until the volume had decreased.

I don't like the trap and would never install anything like this. Not because of the traps bad design though. I could live with using that trap. However I don't like that the exit is about an inch lower than the wash machine discharge. This seems an easy way for a small clog to create backup in the line. But I don't think this is an issue unless this is clogged up.

I also highly doubt it is your water. The easiest way to test this is to get a pail of water and soak clothes that smell as close to "nothing" as possible - so not your damp smell and not the fresh just washed smell. Let the clothes sit for a couple hours in the bucket and see if they smell after they dry. If this is your issue you are going to have to buy some sort of filtration system. But really this would need to be installed close to your tap as you might be stinky from showering due to this (hence the reason I don't think this is your issue - I have showered in well water houses... and you know).

The most logical reason is due to some mold or bacteria buildup in the wash machine. Take out all of the filters and clean them (you may have to read your user guide! the horror). Then run straight bleach for a few cycles, and leave machine open to dry. You may have to do this over a couple days a few times. You may also have to wipe under the seals and open up the back to see if there is anything around the drum. This is not an easy thing to fix other than pouring bleach/chemicals and letting your washer dry out. The repair guy sounded like a flake and should have told you this. He probably didn't want to touch it because he could do a pretty decent job taking apart and cleaning but miss something and you are still mad about a smell.

Everyone that does a decent amount of laundry should always be leaving the wash machine door open to dry faster and also to run an empty load on full/very hot with straight bleach. Your clothes will smell better, your machine will last longer.

We once had bad-smelling clothes after washing, then discovered that it was caused by the washing machine filter, that wasn't flushing away dirt water properly, but running it back into the machine, causing the bad smell.

All we did was clean the wash machine filters, then washed clothes returned to smell properly as washed.

It's wrong the plumping need to plump separate to have ventilation pipe two of them to one drain to a vent pipe and then usually just the washer to a drain to vent pipe not all together to to risky for 1 to back up to another