Nobody takes offense in the sun. The word for it comes from the shade.
Umbrage is from Latin umbra, "shade, shadow." Merriam-Webster traces it through Anglo-French and the Latin umbraticum, which derives from umbra. Etymonline gives the route through Old French ombrage and dates the English word to the early 15th century. Dictionary.com says it entered English in the late 1300s or early 1400s.
The first English meanings were literal: shadow, darkness, shade. Etymonline notes that early uses pointed especially to the shade of tree foliage. Merriam-Webster shows that this sense never fully died. It quotes Charlotte Brontë's 1849 novel Shirley, which describes resting "at the foot of some tree of friendly umbrage."
Figurative senses arrived quickly. Merriam-Webster notes that Shakespeare used the word in Hamlet for a faint likeness of a person, and that the sense of resentment was established by the 17th century. Etymonline records "suspicion that one has been slighted" by the 1610s. It suggests the picture is of being overshadowed by someone else and pushed into obscurity.
The phrase "take umbrage at" is attested by the 1670s, according to Etymonline. It is where the word survives in everyday English. Merriam-Webster defines it as irritation over a slight or insult that may be imagined rather than real. It also lists a fainter sense, a vague hint or a reason for doubt or suspicion, which sits close to the suspicion Etymonline records.
Further back, Etymonline gives a Proto-Indo-European root, andho-, "blind, dark." Merriam-Webster instead points to a Lithuanian cousin, unksmė, "shadow." So the offense in umbrage sits at the end of a very long line of darkness.
English has a modern echo of the same metaphor. Etymonline compares umbrage to the slang "throw shade," recorded by 2013, which also casts a shadow over someone to put them down.