No, the phrase "taking the mick/mickey" comes from the phrase "taking the piss" which in turn has it's origins in the Cockney rhyming slang "taking the Mickey Bliss".....meaning, "taking the piss" aka poking fun at someone or something.
Here:
"The word 'piss' (or, in Scotland, 'pish') has its origins in Latin (pissare) and French (pisser), an onomatopoeic word to describe the sound of urination. One of the mildest swear-words, it nevertheless has a major influence on a number of common British phrases.
When men sleep, the build-up of urine in the bladder puts pressure on the man's prostate gland, resulting in an erection. In the 17th and 18th Centuries, a man who was thought to be unnecessarily arrogant would be described as 'piss-proud'. The New Canting Dictionary of 1725 contained an entry on 'vain-glorious or ostentatious me' which read:
One that boasts without reason, or, as the Canters say, 'pisses more than he drinks'.
If you were to ridicule someone for being too 'full of themself', you would 'take the piss' out of them. As the word 'piss' became categorised as vulgar, the phrase was modified - 'taking the micturations', later shortened to 'taking the mickey' (nothing to do with a person called Michael). With the invention of the urinal, gentlemen would aim towards a small illustration of a bee, intended as a 'pissing point', the Latin word for bee being 'apis', while a domestic commode would be referred to as a 'piss-pot'. In Britain, the link between alcohol and urine is clear: a night out might be described as 'going on the piss'; if a person appears to be spending his money excessively on alcohol he is said to be 'Pissing it against the wall' - presumably on the same principle that you don't buy bad beer, just rent it; a 'piss-up' is a drunken party or pub-crawl."
Another word for micturation is....urination. And micturations was shortened to "mick" as "piss" became a vulgar term.
So, no, the phrase "taking the mick" isn't racist nor does it refer to the Irish as a group. 🤦♂️