I adopted a foal from the adoption centre but when it arrived it was an adult for some reason. I thought maybe the foal aged before it arrived to my house so I shrugged it off. Then, a couple of sim hours later it aged again and became an elder. I don't really understand why because the age bar clearly said it had 53 days until it became an elder.

Does anyone know how to fix this?

I've seen Service/Role Sims do that, but not so much resident sims which is what an adopted pet would become. This might be a "feature" of EA story progression, though. I take you are not using a modded aging manager such as Buzzler's or either of AwesomeMod or NRaas StoryProgression?