Find all words with 3 letters with regex

You should use your match with word boundaries instead of anchors:

\b[a-zA-Z]{3}\b

RegEx Demo

When you use:

^[a-zA-Z]{3}$

It means you want to match a line with exact 3 letters.

you can use . instead of [a-zA-Z] if you want to match any character (also numbers):

 \b.{3}\b