Typically:
The country where you are tax resident will want to tax you on worldwide income; and this according to its local (tax) legislation.
Many countries will want to tax you on the money earned while working from that country.
In many cases of DTT you will get "credit" in your tax-resident country for taxes paid in other countries.
The 90 in 180 constraint is per person, not per passport.
Your entry and leaving is tracked.
Leaving would probably not be a problem.
Coming back (anywhere in Schengen) will be.
Also, you will need to commit perjury on future visa applications for many countries: "have you ever overstayed a visa?"
You want to be become a non-employee, remote collaborator, abroad.
Does your company allow you to these aspects independently? Do they have people working as contractor? Do they have people working remotely? Do they have people working abroad? All of this in a role similar as you?
There are a number of reasons (*) an employer does not want to deal with all consequences of an employee working abroad. As non-employee abroad you will need to deal with yourself; for the countries that you are working from.
If you want them to go along with this, you will need to convince them that you are/will be covering all of this yourself.
Think: contractual agreements as non-employee, right to work, all practical aspects of working 'very' remote (connectivity, availability, accessibility, confidentiality, ... ), taxes ( theirs and yours), contributions ( theirs and yours) for healthcare + social security.
Check your contract and employee handbook for aspects that your company find important.
Many us states/countries will want to tax you for your earnings while working from the state/countries.
Based on a number of criteria some will consider you tax resident. They will want to tax you on your worldwide income. You would typically get 'credit' for taxes already paid elsewhere.
If you are US or Eritrean citizen your country of citizenship also require you to submit a tax return.
Please note that the us and many other countries require a work permit or citizenship to be able to work legally in/from their territories.