Is this happening to anyone else? I just brought a couple of games on sale and I have the receipts so I know the transactions went through but they don't appear anywhere in my "owned games".

Owned Games as in the section in GOG Galaxy? That can take a while (perhaps a day) to update for new purchases.

You can accelerate the process by completely closing the client and restarting it. Re-syncing with your GOG account will take place once it launches.