c. 1200, pece, "fixed amount, measure, portion;" c. 1300, "fragment of an object, bit of a whole, slice of meat; separate fragment, section, or part," from Old French piece "piece, bit portion; item; coin" (12c.), from Vulgar Latin pettia, probably from Gaulish pettsi (compare Welsh peth "thing," Breton pez "piece, a little"), perhaps from an Old Celtic base kwezd-i-, from PIE root kwezd- "a part, piece" (source also of Russian chast' "part").
So yes, it looks like coincidence.
(Peso is a word in Spanish and up until modern times, there's very little transfer from English to Spanish)
Part of the reason for this is that English didn't even exist until fairly recently (though still before modern times). But there were various Germanic languages, some of which English evolved from. There are a fair number of Spanish words of Germanic origin, some fairly old, perhaps dating from the Visigothic kingdoms in Iberia: guerra, guante, rico, orgullo, arpa, blanco, banco, frasco, yelmo, flecha, emboscar, jardín, flotar, bisonte, etc.
More specifically with respect to English, though, https://en.wikipedia.org/wiki/List_of_Spanish_words_of_Germa... says that bote comes from Old English "bāt" via Middle English "boot" and then Old French "bot". The other words in that category include arlequín, este, norte, oeste, and sur/sud-. But when did those words make the jump into Spanish? Some, like arlequín, were fairly recent!
"Old French" was supposedly spoken up to the mid-14th century, so words that came into Spanish directly from Old French probably came in before 01375 CE. But there's always the possibility that the word lingered for a century or three in some intermediate dialect like Occitan or Catalan before making it into Spanish. Unfortunately I don't know of any resource in Spanish comparable to the OED or Etymonline to find old Spanish attestations.
The phonetics would have been even more similar at the time, but I suspect it's a coincidence. The words stem from different origins†, the sense development of "piece" is straightforward (a coin was a chunk of precious metal cut off the end of an ingot, thus being a piece of that ingot, and struck), and the chronology is probably wrong for a phonetic influence. https://www.etymonline.com/word/piece says "piece" for a coin is "c. 1400", which would be about 200 years older than the earliest attested occurrence of "piece of eight" and 100 years older than the peso itself, which was introduced in 01497.
However, the earliest attestation given in https://archive.org/details/oed07arch/page/836/mode/1up?view... is from 01575, in Scots: "To be payit all in half mark pecis," and the first attestation of "piece of eight" is from 01610: "Round trunkes, Furnish'd with pistolets, and pieces of eight." Maybe Etymonline knows of a much earlier attestation? Because if "piece" in the sense of "coin" really didn't come into use until the late 16th century, a phonetic influence would be much more plausible.
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† "Peso", like "poise", is from Latin "pensum", "weight", which may be from "pensare", the frequentative of "pendere", "to hang, to weigh", while "piece" comes from Latin "petia" or "pecia", "fragment". In Spanish today "peso" is still the normal word for "weight" (and "pesar" is the normal verb for "weigh") and "pieza" is still a fairly common word for "fragment".
(From) latin, pensum.
Piece, from https://www.etymonline.com/word/piece#etymonline_v_14956
c. 1200, pece, "fixed amount, measure, portion;" c. 1300, "fragment of an object, bit of a whole, slice of meat; separate fragment, section, or part," from Old French piece "piece, bit portion; item; coin" (12c.), from Vulgar Latin pettia, probably from Gaulish pettsi (compare Welsh peth "thing," Breton pez "piece, a little"), perhaps from an Old Celtic base kwezd-i-, from PIE root kwezd- "a part, piece" (source also of Russian chast' "part").
So yes, it looks like coincidence.
(Peso is a word in Spanish and up until modern times, there's very little transfer from English to Spanish)