Some countries don't allow dual citizenship at all, that is, if you carry a passport of another country, you can't remain a citizen of the original country. The USA for instance requires that you give up your US citizenship if you are naturalized in another country.
Nothing you said is true. I know because I hold 3 citizenships, USA, PL, and DE.
Poland and the USA allow dual or multiple citizenships, and I was naturalized as a Polish citizen after being a US citizen, and later recieved my german citizenship. I hold all 3 citizenships and never had any problems.
And you CAN enter Poland on a US passport even if you hold a polish passport, but why would you, since your American passport would get stamped with a schengen visa limited to what, 90 days? So convex, why would you spout ******** like this?