You're seriously quoting Wikipedia as proof of something? Have you seen the talk page on that article? There's no reference to a source for that formatting of e.g.2Q2016. The authors are debating the definitions themselves!
As for Investopedia... If I prove to you that the description of Quarters on that website in relation to the filing of the Securities and Exchange Commission form 10-Q is incorrect, will you concede that the MySQL documentation should include a more complete and descriptive statement of the expected return values from the quarter function?
IF Investopedia's article were correct, then there should be no way that a company can file a form 10-Q with a quarterly end date of November 30th, correct? Yet ORACLE itself filed a form 10-Q with that very date:
One can look in "Fundamental Accounting Principles." by Kermit D. Larson. I have the 1996 edition. It states that "the accounting process is based on the time period principle... The annual reporting period is not always the same as the calendar year ending December 31. In fact an organisation can adopt a fiscal year consisting of any 12 consecutive months. An acceptable variation of this is to adopt an annual reporting period of 52 weeks."
As for countries where it makes a difference, how about the USA? Under US GAAP (Generally Accepted Accounting Practices, a hefty tome published every year or so which defines how accountants are expected to operate and which is produced in order to standardise accounting practice within a legally regulated nation state and ensure that accountants produce legally compliant documentation) a subsidiary company can have a fiscal year which varies by up to three months from the parent company, and can file returns with that variance. Under International Financial Reporting Standards, a system of GAAP adopted by hundreds of countries, a subsidiary company may also have a different financial year to its parent company, but if they operate in that fashion must produce two sets of accounts, with one set aligned to the parent company.
Many countries have "New Year's Day" on a date other than January 1st; China, Cambodia, Thailand, Ethiopia etc. Though most countries concede to use a variation on the Gregorian system for reasons of International trade and travel. I accept that MySQL documentation does state it uses the ISO 8601:2004 system, but that Standard does NOT contain a definition of a quarter.
I note that the documentation for the Java Platform SE 8 IsoField class which, although still debatable, does at least make it clear how the class operates.
I have submitted a "bug report" feature request for an expansion of the QUARTER() function to include modes and/or year start dates. Being new to the backroom of MySQL, I was unaware that documentation issues should be directed to the Bugs forum instead of the (what I thought would be "common sense") forum entitled "Documentation". But there you go! Maybe "common sense" isn't so obvious after all.