wordpress category page not showing wrong posts

The category about animal names is showing posts about essays because you have added the posts about essays in the “animal names” category.

Perhaps the “animal names” category is your DEFAULT category which is automatically selected, and you didn’t take care to uncheck it when you made these posts.

WordPress doesn’t care what names you give your categories: you have to put the appropriate posts into the appropriate categories yourself. If you call a category ANIMALS but put a post about MANGO into this ANIMALS category, WordPress will do exactly that for you: your post about MANGO will be listed in your ANIMALS category archive page on your website.

So log on to your WordPress dashboard, check the affected posts, and confirm that these posts have the appropriate category.