Translation:My father is a taxi driver.
It's not just redundant, in most cases it's rude! The reason you use 父 for your own father and お父さん for other people's fathers is that the latter is an honorific form to show respect. Using the honorific form while speaking to others indicates that you believe the listener should also show your father that respect, and due to Japanese's in-group/out-group mentality, also implies that your group (i.e. your family) is socially situatated above the listener. For that reason, お父さん should generally only be used while speaking directly to your father, or to somebody who shares the same in-group as you and your father (e.g. your siblings or your mother).
The question I don't understand is, why is the 私の necessary? If you were talking to your siblings or your mother, whose father would already be implied by context.
How about 運転車? Self-driving car? (Apparently so.) I find it amusing that a car-driver and a car driver sound the same.
It seems that 手 means "hand" and 者 means "person". What that tells us about the implications of 運転手 versus 運転者、I can't say.
I'll be on the lookout for an 運転猫、though.
Keep in mind that "hand" in English can mean "An agent; a servant, or manual labourer, especially in compounds; a workman, trained or competent for special service or duty; a performer more or less skilful." We have words like "deckhand" which is a person who works on a ship.
In Japanese, 手 can work the same way.
It turns out self-driving car is called 自走車 (jisōsha) in Japanese :)
手 comes from Chinese, and it's indeed interesting why some professions are referred to as ~手 instead of ~者, e.g. 射手 (archer), 歌手 (singer), 水手 (sailor, not used in Japanese), etc. None of them seem to be associated with the traditional "upper class" thogh.
I didn't find 運転者, but I stumbled across this at Wikipedia. (Sorry, no English page available.)
Executive summary: Variants include 運転手, 運転士, ドライバー, 機関士, 操縦士, オペレーター, 車夫, 駕篭かき...
旅客輸送を職業とし、ドライバーやプロドライバーとも呼ばれる。自家用の貨物自動車や乗用車、自家用バスを運転する者も運転手と呼ばれる。 これに対し、（自動車運転免許以外の）資格が必要な鉄道や軌道の電車・汽車、船舶や航空機、または機械を運転する職業は機関士や運転士、操縦士と呼ばれる。鉄道や軌道では、機関士や運転士と呼ばれる（動力車操縦者を参照）。一部の鉄道・軌道業者では運転手を社内呼称として用いる業者もある。 クレーンなどの操縦者は、運転士あるいはオペレーターと呼ばれる。 旅客輸送に従事するかつての職業には、人力車の「車夫」や、駕篭をかつぐ「駕篭かき」などもあったが、現在では観光向けのものがほとんどである。
Can somebody think of a situation where this construction would be appropriate and not rude? If you were talking to someone in your immediate family, 私の would already be implied, and would probably know what your dad does for a living. A parent talking to their child? (Though I would imagine おじいさん more appropriate in that case.) A cousin maybe? I'm not sure how far the concept of family in-group goes for the purpose of words like お父さん and お母さん.
For fun, I ran my (accepted) version and theirs past SpanishDict and Giggle Translate.
• My father drives a taxi.
• My father is a taxi driver.
SD didn't insert the definite articles that I half-expected, but conflated the two with a single translation, making me feel vindicated.
Mi padre conduce un taxi.
GT distinguished between the two.
Now where have I seen the first one before? Could it be that the course providers are supplementing their junior high textbooks with the even less reliable GT?