Britons have been using this figure of speech for decades, if not centuries. A “Mickey” of course, is a “Mick”: a pejorative, racist term for an Irishman (so nicknamed because so many Irish surnames begin with Mc- or Mac-) It is a common stereotype, in both the UK and USA, that Irish men have volatile tempers, like to brawl, and make good boxers. So, To “take the Mickey (out of someone)” means to take the fight, the vigor, the gravity, the self-importance out of them, by mocking them, usually in a very subtle way.

Headmaster: “…so I expect you boys to comport yourself with the full dignity befitting students of this establishment of secondary learning.”

Student: “Oh yes, we will sir. We’ll even wear our school blazers to bed.”

Headmaster: “If I didn’t know better, I’d think you were trying to take the Mickey out of me!”

RE:
What does the phrase "taking the mickey out of me" mean?

vb. British —

meaning to mock, deride, poke fun at.

These expressions are milder versions of ‘take the piss.’ Unbeknownst to most users, they employ rhyming slang: Mickey is short for a mythical ‘Mickey Bliss,’ providing the rhyme for ‘piss.’ ‘Michael’ is a humorous variant. The phrases, like their more vulgar counterpart, have been in use since the 1940s.”

take the mickey (out of someone) Vrb phrs. To tease, to ridicule. Also shortened to take the mick. An abbreviated form of the Cockney rhyming slang take the mickey bliss, meaning ‘take the piss’. E.g.”Stop taking the mickey out of Billy, he’s very sensitive and you’re upsetting him.” Cf. ‘take the Michael’ and ‘extract the Michael’. [1930s]