I was surprised at the link between these two. From OED:
Caribbean - possibly from Carib of which OED says:
Etymology: < Spanish caribe, adjective and noun (late 15th cent.) < a Proto-Cariban base with the sense ‘man, human being, indigenous person’, probably originally transmitted via an Arawakan language. Compare Middle French, French †caribe , noun (1568). In γ. forms after French caraïbe, adjective and noun (1658). Compare slightly earlier cannibal n.1 and the discussion at that entry.
Compare also Garifuna n.
The origin of the French form with -aï- is unexplained; compare Spanish †caraibe (c1550 in an apparently isolated attestation with reference to Chile).
Some of the variants with final -e may reflect the trisyllabic pronunciation of the Spanish etymon; compare Caribbee n.
The β. forms reflect the pronunciation of intervocalic -b- in Spanish.
The Caribbean was apparently originally inhabited by peoples speaking Arawakan languages, while Cariban languages were spoken in the north-east of South America. Speakers of Carib then expanded aggressively into the Antilles region (prior to contact with Europe), which led to the development of a new language known as Island Carib, which is structurally Arawakan, but shows strong Carib influence in its vocabulary. This is the language described by R. Breton in French as caraïbe in the 17th cent. Similarly, English Carib and related words are also sometimes used to refer to the Island Carib language and people (compare e.g. quots. 1666 at Caribbean adj. and 1807 at Caribbee adj.). The Black Carib (Garifuna) language is a form of Island Carib. For a more detailed discussion compare N. L. Whitehead Arawak Linguistic & Cultural Identity through Time, in J. D. Hill & F. Santos-Granero Comparative Arawakan Histories (2002) 51–73.Originally: a member of an indigenous Central and South American people inhabiting the Lesser Antilles and neighbouring mainland coastal regions at the time of the arrival of Columbus; (subsequently) a descendant of these, now inhabiting mainly coastal regions of French Guiana, Suriname, Guyana, and Venezuela, and on some Antillean islands, such as Dominica and Trinidad. Also in early use: †any member of an indigenous people; esp. a Central or South American; (more generally) a barbaric native, a savage (often formerly equated esp. by Europeans with warlike and savage behaviour, esp. cannibalism (cf. cannibal n.1 1a)) (obs.). Cf. Caribbean adj.
and OED on cannibal:
In 16th cent. plural Canibales, < Spanish Canibales, originally one of the forms of the ethnic name Carib or Caribes, a fierce nation of the West Indies, who are recorded to have been anthropophagi, and from whom the name was subsequently extended as a descriptive term.
Professor J. H. Trumbull, of Hartford, has pointed out that l , n , r interchange dialectally in American languages, whence the variant forms Caniba , Caribe , Galibi : and that Columbus’s first representation of the name as he heard it from the Cubans was Canibales , explained as ‘los de Caniba or Canima’; when he landed on Hayti, he heard the name of the people as Caribes and their country Carib ; the latter was afterwards identified with Puerto Rico, named by the Spaniards ‘Isla de Carib’, ‘which in some islands’, Columbus says, ‘they call Caniba , but in Hayti Carib ’. Apparently, however, it was only foreigners who made a place-name out of that of the people: according to Oviedo ( Hist. Gen. ii. viii.) caribe signifies ‘brave and daring’, with which Prof. Trumbull compares the Tupi caryba ‘superior man, hero, vir ’. Calib- (in Caliban n.) is apparently another variant = carib-an; compare Galibi above-mentioned.
Columbus’s notion on hearing of Caniba was to associate the name with the Grand Khan, whose dominions he believed to be not far distant; he held ‘que Caniba no es otra cosa sino la gente del Gran Can’. To connect the name with Spanish can, Italian cane, Latin canis dog, was a later delusion, entertained by Geraldini, Bp. of San Domingo, 1521–5; it naturally tickled the etymological fancy of the 16th cent., and may have helped to perpetuate the particular form canibal in association with the sense anthropophagi. See Prof. Trumbull’s article, in N. & Q. 5th Ser. IV. 171.
1. Thesaurus »
a. Orig.: †a name given to the Carib people of the West Indies, who were said to eat human flesh (obs.). In (later) extended use: a person who eats human flesh; = man-eater n. 1.
1553 R. Eden tr. S. Münster Treat. Newe India sig. Gviijv, Columbus..sayled toward ye South, & at ye length came to the Ilandes of the Canibals.
The name Caliban is a variant of cannibal or from Carib.