I have the exact same issue! I've had it for years and always thought it was just a strange querk I had. Whenever I say little phrases, I constantly have to say something right afterwards that rhymes. Usually the words make sense, but sometimes they are made-up words. It's never really bothered me and my family laughs at some of the silly things that I say, and so do I. Some examples when talking to my dog are:
Walk walk walk, chicken squawk
Let's go, smoochy snow
Dinner dinner! Licky liver
Pretty girlie, chicken squirly
I have NO idea why most of them contain the word "chicken" (whether saying something to my dog or not), lol.
The rhyming only happens with short exclamation-type phrases.