The etymology is contested. It is common sense that Anglican derives from anglo-, an archaic form of eng- as in England, and scholarly concensus holds, that the ethonym Angels derives from a toponym known from Latin sources as Anglia (German Angeln, Danish Angel). However, sources on the original meaning of the name differ in their presentation of the evidence and have in fact very little to recommend themselves.
The interpretation of an ancient placename is necessarily up to speculation, not the least because the pre-history of Germanic tribes is poorly documented. The Proto-Indo-European reconstruction fares no better, I'm afraid. Notabene: "In der Onomastik sehen viele geradezu ein Paradefeld der Volksetymology" (Fetzer, Aspekte toponymischer Volksetymologie, 2011: 59. with further references, emphasis mine). For example, the name of Angermünde was transfered from Tangermünde, which was frequently spelled without T- in the middle ages (Niemeyer, Deutsches Ortsnamenbuch, 2012).
My favoured interpretation in this answer would be that German Enge is cognate, whence also Angst, compare jmd. in die Enge treiben. I do not remember who said it, that the word could describe a narrow fjord.
On a website of the Saxon Academy of Sciences I found further details on the etymology of engi, ango and angul: "angul m. a-St. ‚(Angel)Haken, hamus‘; erst vom 11. Jh. an übernimmt angul die Bed. von
ango¹: ‚Stachel, Spitze, aculeus, fuscina, acumen‘; ‚Türangel‘, auch ‚Endpunkt, cardo, finis‘."
In fact, their roots appear to differ:
engi "narrow":
Am nächsten verwandt sind außergerm. u-Stämme (zu idg. *ang̑hu-): [altindisch] aṁhú- ‚eng‘; [lateinisch] angi-portus (< *angu-) ‚enges Gäßchen‘;
angul "hook",
(dazu vielleicht der VN Angul-[Ongol-]s[e]a-xan)
[...] Dem Ansatz von idg. *ankúlos entspricht in
Form und Bed. gr. ἀγκύλος ‚gebogen, krumm‘;
dazu kommt gr. ἀγκύλη ‚Schlinge am Wurf-
spieß, Riemen; Haken, Türangel‘,
I do note that Egyptologist Philippe Collombert 2023, "The Egyptian Hieroglyphic Sign for the Sky 𓇯", has argued that a door-hinge was the original motivation of the sky hieroglyph. If this is correct, we might as well compare (*h)angel to Himmel < *h₂ek̂- ~ h₂k̂-* "sharp". But this does not seem very likely to begin with since heaven and Himmel are notoriously uncertain.