The plain part of the story is true: red tape was once real tape, and it was red.
Etymonline says the phrase refers to red tape used in Great Britain and the American colonies to bundle legal and other official papers. It notes that such tape is mentioned from the 1690s. Word Histories and wordorigins.org find an earlier literal use. It is a lost-papers notice in The Publick Intelligencer of December 6, 1658, for "a little bundle of Papers, tied with a red Tape." Merriam-Webster gives the same explanation and dates the current meaning to 1736.
That date comes from a poem. Merriam-Webster, wordorigins.org and Word Histories all point to John Hervey's Poetical Epistle to the Queen of 1736, which contains the line "Red tape and wisdom at the Council show." The tape here already stands for official routine.
Red tape stayed a physical object for a long time. In Bleak House in 1853, Dickens lists "red tape and green ferret" among the goods of a law stationer, which is still the literal tape. An 1861 note in Notes and Queries, quoted by wordorigins.org, called the phrase "probably of no great antiquity."
The popular origin story is the one worth treating with care. It says that Charles V, Holy Roman Emperor and King of Spain, had his ministers tie urgent papers with red ribbon to separate them from ordinary files. Some versions add that Henry VIII did the same with petitions to the Pope in 1527.
The sources checked here do not support those claims. One popular history article states the Charles V account as fact but cites no contemporary document, and admits it is "not clear" that Henry VIII's court called it red tape. The earliest evidence the dictionaries rely on is English and later: tape in the 1690s, and the figurative phrase in 1736.
So the safe version is short. Red tape bound official papers in the English-speaking world, and by the 1730s writers were using it as a figure for officialdom. A Spanish imperial origin may be true, but nobody has shown it.
Etymonline offers a close parallel. Middle English green wax meant a notice of a fine, from the green wax used on the Exchequer's seal.