A company could see you are working at multiple places, but the tax authorities could say you can't work multiple full time jobs maybe? In the us I bet you could just do it. You'd have opt out of those horrible places where companies want to share your salary info. I really can't believe that is actually legal still, even though I know it is since my own salary history was mostly there. Little companies like startups I have worked in didn't do that.
Tax authorities never prevent you from doing it. You might have to file some more forms though.
In Australia employees provide a form to their employer which is basically "set aside tax from my pay as though this is my only source of income" vs "tax me at a higher rate".
IIRC the wording on the form is simplified to make it seem like the question is "do you have another source of income?"
That makes sense. In the USA, we can just specify additional withholding without any justification. This can be the result of other income (stocks included), but also due to people having a working spouse.