Q: Does the meaning of angle here stems from the geometrical figure, or from the archaic sense of hook listed in the Oxford Dictionary?
It's short for angle worm, which gets its name from the fishing hook.
The OED defines angleworm as:
An earthworm of a kind used for bait by anglers.
They say it's chiefly US and the author Guy Wetmore Carryl was also American. His poem was published in 1898 and the OED has quotations from 1788 onwards, and the 18th and 19th centuries are of the form angle-worm or angle worm.
The OED says the angle of angle worm originally was a fish hook and goes way, way back to early Old English, with cognates in Germanic languages and etymologically is:
ultimately < the same Indo-European base as ancient Greek ἀγκύλος crooked, curved (see further angle n.2 [geometrical etc.]).
How about the geometrical angle? Well, it's related to the fishing hook, but the common ancestors are a long way back. The OED says the first use of this word was a corner, recess or nook of a room or enclosed space; used this way before 1325. It also meant a point of land; a cape, or promontory; used before 1387. Other geometrical meanings are from before 1398.
Etymologically these came to English through Anglo-Norman, Old French, Middle French,
ultimately showing a variant of the same Indo-European base as ancient Greek ἀγκών bend of the arm, nook, bend, angle, ἀγκύλος crooked, curved, and angle n.1 [the fishing hook]
Finally, the names of England and English come from the Germanic tribe Angles, which may derive from a place called Angeln (on the German coast near the Danish border) which itself may come from the fishing hook.