I was under the impression that we can only adopt pets once a day, so I waited and the next day when I went to take a look it said:

Oops - The Following Errors Occured

• You already used the Pound services today.

 

but I hadn't use the services yet. I would like to but I don't know what the problem is. Was I wrong about the amount of time between using the pound? Or is there a bug? etc.?

It is once per game day. Say for example you adopted a pet from the pound at 9 pm at night. Then the next morning you wake up and try to adopt another pet. The above error will occur because while it may be a new day for you, it is the same day in the game.

try again in 5 minutes and the error should be gone.

Look at the game time (right above your username in the top right corner). If you adopted a pet at 23:55, you can adopt one again at 1:00, five minutes later. Once a day means once every set of 24 hours in game time.