Given it's unclear on the actual relationship between member worlds, who's to say, that the Federation is actually based on a European Union model of governance, not, the American federal model, if in fact it's the former, having an ambassador on other member worlds makes sense.
When the European Union establishes a single legislature and a single popularly elected president and a single cabinet, all of which is NOT comprised of or controlled by officials from the Member state governments; when it starts conducting a single foreign policy on behalf of its Members, including having the power to declare and wage war without its Member governments' permissions; when it establishes a single fiscal policy and common currency for all of its Members; when it establishes its own standing army; and when its President gains the power to declare martial law and place its standing army on the streets of its Members...
It's not obvious that the Federation has exclusive competence in many of these areas. "Reunification" depicts Vulcan as having standing military forces,
The State of Ohio maintains a standing military force, including its own naval patrol. This does not mean the State of Ohio is a sovereign state.
while despite being a world in a federation with a moneyless economy the Bolian homeworld has its famous bank.
Star Trek is full of contradictory information about whether or not the Federation is truly "moneyless," including contradictions about whether or not Earth itself is moneyless.
The idea of the Federation as a polity where the central state dominates and the member-states have only residual powers, on the model of American federalism, doesn't strike me as intuitively plausible.
It's perfectly fair to argue that the Federation's version of federalism involves greater autonomy for the Member worlds than American federalism currently involves for U.S. states, but the preponderance of evidence is that the Federation is clearly a sovereign state in its own right -- far moreso than the current European Union.