The English language is more than just a collection of words, it entails grammar as well. "Their" is plural. For singular, it's "his," "her," "its."
For example, the sentence, "Pat has a flower. Their flower smells good, " doesn't sound right because "their" is plural. "Pat has a flower. Its flower smells good, " doesn't sound right either because "its" is reserved for things and not people.
Also, the dictionary isn't a good source for grammar. It contains definitions of words, proper or not, like "ain't" and "irregardless."
https://www.merriam-webster.com/dictionary/irregardless?utm_...
https://www.merriam-webster.com/dictionary/aint
