what does "by the middle of the week" mean?

It is a vague limit, is what it is.  What it means is vague as a result.  We do not know for certain what the person considers the middle of the week.  Is it Wednesday as a general thing, Wednesday at noon specifically, before Wednesday is done, or before Wednesday started?  For me, "middle of the week" would usually be understood as Wednesday, and if I wanted to be certain that I was done before the "middle of the week", I would make sure to be done before Wednesday came around.

Now, if it had been "by the middle of the week", I would generally think that Wednesday is part of the time period allowed for finishing.

"By" tends to be inclusive.  Before tends to be exclusive.

But I wouldn't be all that happy with having someone give me such vague instructions when there might be penalties associated with failure to meet those instructions.  I would ask for clarification.  It is not very equitable to penalize when you were not told clearly what your real obligation was.  If it is that important, the person ought to be pretty clear.  If they aren't, it cannot be that important.

Wednesday is the middle of the work/school week, yes. and 'by then' usually means 'on or before' -- if the speaker had meant Tuesday at the latest, his/her language would be different

I use a lot of estimates at work.  Like middle of the week.  IF I'm being specific I'd say Wednesday.  Probably even Wednesday afternoon.

But saying middle of the week I'd mean late Tuesday, Wednesday or Thursday.  

You’re overthinking. It’s roughly Wednesday-ish. The middle of the week has some flexibility.

It's an approximation. It means it could happen on Tuesday, Wednesday or Thursday.