My first answer to this riddle suggested that there was a common word hidden directly or indirectly in each line. That turned out to be wrong, but I've kept the answer below.
The question has been updated. In the light of the update, I think ...
... that the two lines of each stanza describe homophones. The hint that the dialect might break the puzzle means that pronunciation is important and that probably means that we are looking either for homophones or for rhyming pairs.
I've found a pair for each stanza, some more convincing than others. But the last pair makes me confident that what we are looking for here are homophones. Oh, and the title hints at homophones often being confused with each other.
Well, here goes:
Sometimes it's an alternative
Sometimes it belongs to the cow
other / udder. Despite the the ð/d difference, this seems to be a reasonable homophone pair.
One day it's orange
Another day, it's not pure somehow
carrot / carat. Not quite sure why carat should be "not quite pure", but it's a measure of the purity of gold.
Sometimes it's the important date
Sometimes it's only in the morning
due / dew.
One day it needs lenses
Another day, it is worth seeing
sights / sites. This seems a reasonable guess, although both words should be spelled "sights" in my opinion. (The sights, i.e. the aiming device on a rifle, are optical and therefore need lenses.)
Sometimes it wakes up
Sometimes it cries
morning / mourning. (Thanks for the gentle nudge from OP in the comments.)
One day he can answer this
But what is that, under your eyes?
knows / nose.
This is my initial answer to the riddle. It turned out to be wrong.
But what is that, under your eyes?
It is OR, which is either hidden directly in each line or hidden in something that line describes.
Sometimes it's an alternative — or
Sometimes it belongs to the cow — horn
One day it's orange
Another day, it's not pure somehow — immoral
Sometimes it's the important date
Sometimes it's only in the morning
One day it needs lenses — shortsighted
Another day, it is worth seeing — portrait
Sometimes it wakes up — thunderstorm
Sometimes it cries — newborn
One day he can answer this — word
But what is that, under your eyes?
And the title?
In computing, bit-wise OR is a binary operator that sets the bits of the result when any of the respective bits of the input is set.
