early 15c., "shadow, darkness, shade," from Old French ombrage "shade, shadow," from noun use of Latin umbraticum "of or pertaining to shade; being in retirement," neuter of umbraticus "of or pertaining to shade," from umbra "shade, shadow," from PIE root *andho- "blind; dark" (source also of Sanskrit andha-, Avestan anda- "blind, dark").
The English word had many figurative uses in 17c.; the one remaining, "suspicion that one has been slighted," is recorded by 1610s; hence phrase to take umbrage at, attested from 1670s. Perhaps the sense notion is similar to whatever inspired the modern (by 2013) slang verbal phrase throw shade "(subtly) insult (something or someone)."
early 14c., from Anglo-French esquirel, Old French escurueil "squirrel; squirrel fur" (Modern French écureuil), from Vulgar Latin *scuriolus, diminutive of *scurius "squirrel," variant of Latin sciurus, from Greek skiouros "a squirrel," literally "shadow-tailed," from skia "shadow" (see Ascians) + oura "tail," from PIE root *ors- "buttocks, backside" (see arse). Perhaps the original notion is "that which makes a shade with its tail," but Beekes writes that this "looks like a folk etymology rather than a serious explanation." The Old English word was acweorna, which survived into Middle English as aquerne.
also Ben-Day, by 1905, a printing and photoengraving technique involving overlay sheets of small dots or lines, used to create shadow effect, etc., named for U.S. printer and illustrator Benjamin Day Jr., who developed it c. 1879.
"cat," by 1690s, a diminutive of puss (n.1), also used of a rabbit (1715). As a term of endearment for a girl or woman, from 1580s (also used of effeminate men), and applied childishly to anything soft and furry. To play pussy was World War II RAF slang for "take advantage of cloud cover, jumping from cloud to cloud to shadow a potential victim or avoid recognition."