I would not say it is "normal", but it does happen. I suppose it depends on the circumstances though such as: where are you selling vs buying, do you know more than one Realtor, were you happy with the services that were provided (either from the listing Realtor or Buyer's Realtor). Hope that helps!
It's certainly not unheard of or unique to do that. But usually home owners choose a realtor to sell their whom who they feel confident enough in to also help them purchase a home. It also helps the process along to only have one realtor as it can be a tricky timeline of selling one home and buying another.