Freezing and thawing process do damage exosomes. After the extraction is better to keep them at -80ºC for long periods of time (above months) at -20 for a few weeks and at 4ºC if you're planning on using them during the week.
Here is a paper I recommend for that question
Article Effect of pH, temperature and freezing-thawing on quantity c...
exosomes are pretty durable entities, so single freeze/thaw cycle does not hurt them much. And also in some cases we receive frozen plasma or serum samples- for which we cant control harvesting/storage/transportation (ideally of course you want to process everything fresh or stored at 4C for 1-2 days)