The full question

Can research for design sometimes look like Qualitative Research for market and social research? Do service designers ever do a bit of market research?

My two cents

Indeed, qualitative research looks pretty similar to market research, social research or Service Design. A few details might change here and there, but basically, it's 99% the same.
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Where service design does differ in the 1% is:

• It's looking for actionable things to change in the service

• It's not only focused on the end user but also takes into account the people who make the service happen (employees, stakeholders, partners)

So yes, market research is another tool that service designers can use.