To call a horse a Gee Gee, which is usually used when speaking to small children, comes from the instructions from a farmer to his working horses, (usually Shire or Heavy Horses). The farmer did and still does today, say, "Gee-up" when he wants the horse to move forward, and "Whoa" or "Whoa-up" when he wants the horse to stop.
Hope this helps. Schutz.
There is no record of the word gee-gee being used to mean a horse prior to the latter half of the 19th century. So - if it had anything whatever to do with Chester Racecourse 400 years earlier - it certainly took a long time to catch on!
An early use of the verb, gee, meant to go, so that may possibly explain how it came to be an instruction to a horse...and from there to a name for a horse.