"Today is Monday."
Translation:Dziś jest poniedziałek.
The word 'to' can join two nouns, and dzisiaj is an adverb. So we use jest instead
Should the object not be in the instrumental? We're saying that one thing is another thing.
Yes, but if you imagine this sentence as "X is Y", the Instrumental rule only works if both X and Y are noun phrases. "dzisiaj"/"dziś" are adverbs.
See the link I posted above.