I would say it isn't addressed by our nationality laws rather than it isn't recognized in U.S.
SCOTUS recognized the concept of dual nationality in Kawakita v. United States.
From the State Department's Foreign Affairs manual:
7 FAM 082 DUAL NATIONALITY AND U.S. LAW -- GENERALLY(CT:CON-106; 06-06-2005)
Current U.S. nationality laws do not explicitly address dual nationality, but the U.S. Supreme Court has stated that dual nationality is a “status long recognized in the law” and that “a person may have and exercise rights of nationality in two countries and be subject to the responsibilities of both.” See Kawakita v. United States, 343 U.S. 717 (1952).
7 FAM 081 SUMMARY(CT:CON-106; 06-06-2005)
e. U.S. Policy on Dual Nationality: While recognizing the existence of dual nationality, the U.S. Government does not encourage it as a matter of policy because of the problems it may cause. Dual nationality may hamper efforts by the U.S. Government to provide diplomatic and consular protection to individuals overseas. When a U.S. citizen is in the other country of their dual nationality, that country has a predominant claim on the person. A foreign country might claim you as a citizen of that country if (a) you were born there; (b) your parent or parents (and sometimes grandparents) are or were citizens of that country or (c) you are a naturalized U.S. citizen but are still considered a citizen under that country's laws. (The oath you take when you are naturalized as a U.S. citizen (8 CFR 337.1) doesn’t mean the foreign country does not still regard you as a citizen of that country.) Public inquiries about the citizenship laws of other countries should be directed to the embassy or consulate of that country in the United States. 8 U.S.C. 1185(b) (Section 215(b) INA) and 22 CFR 53.1 require that U.S. citizens exit and enter the United States on a U.S. passport, with certain limited exceptions (22 CFR 53.2).
I said specifically that dual citizenship is not OFFICIALLY recognized.
You left off the word “OFFICIALLY.” What is meant by that is dual citizenship is not legal in the US because it is not in our laws as an officially recognized citizenship status.
That snip you posted backs this up:
“Current U.S. nationality laws do not explicitly address dual nationality, but the U.S. Supreme Court has stated that dual nationality is a “status long recognized in the law”
Long recognized in international law. Not so specifically in US nationality laws.
Dual citizenship gives a person rights of nationality in more than one country. *Dual allegiance* ( as in obamas’ case), permitting a foreigner to the Presidency was what John Jay warned against, and why the original wording of the qualifications to be President were changed to what they are now and have never been amended since being signed.