Financial institutions are highly regulated entities. The rules differ based upon the incorporating charter of what sort of financial institution, it is to be.
If you jump through the "bank hoops" you can call yourself a bank. If you jump through the "credit union hoops" or the "savings & loan hoops", you can call yourself one of those.
Who holds the hoops? An agent of the government empowered by legislation to make sure all the i's are dotted and the t's are crossed. Bank regulators may also be credit union, or savings & loan regulators, but don't count on it.
Regulators work within a legal framework defined by federal legislation, state, and sometimes county (see Cook County, IL) legislation. And don't forget case law. There are companies whose sole reason for existence is keeping financial institutions informed about where the legal boundaries lie. And where they are likely to be moved. (I worked for one.)
Years ago a software company decided it wanted the name "SoftBank" or something with "bank" in its name. A distracted idiot would not confuse this company for a financial institution. Nevertheless, regulators came calling and they had to change their name.
Hence, "banking" is a term of art that is defined by law, not dictionaries.
If you have a financial institution that does banking-like functions, it will offer two sorts of products: deposit and lending products. The funds realized from deposit products are used to underwrite lending products. The profits from lending products (i.e. interest) are used to satisfy the contractual obligations of deposit products (i.e. interest).
(If the financial institution is a credit union, they use different words for the same things (e.g dividends). If the financial institution is a savings & loan, it probably went out of business in the late 1980s. If the financial institution is an Islamic bank, now that's interesting since Moslems cannot charge interest.)