Two claims on the same income are the normal case, not the exception. Consider a US-Canadian dual citizen living in Toronto. Canada taxes them because they live in Canada; the United States taxes them because they are a US citizen. Both countries look at the same salary, the same investment income, the same capital gains. Nothing about the arrangement is a loophole or an error, it is simply the intersection of a residence-based system and a citizenship-based one, and it is why the relief mechanisms carry so much weight for this population.
The relief comes from three tools, and they cover different income. The foreign earned income exclusion removes a capped amount of foreign wages or self-employment income from US gross income. The foreign tax credit offsets US tax with income tax already paid to the other country, dollar for dollar, and it reaches investment income and other categories the exclusion does not. An income tax treaty, where one exists, can reduce withholding and assign taxing rights, though its usefulness to a dual citizen is limited by the treaty's saving clause, which lets the United States tax its own citizens as if the treaty did not exist. In many cases the credit alone eliminates the US tax, because the other country's rates are higher than the US rates on the same income.
Filing survives the relief. A dual citizen who owes no US tax after the exclusion and the credit still must file a US return to claim them, and must separately report foreign bank and financial accounts and, above higher thresholds, specified foreign financial assets. These reporting duties apply on the basis of status, so a zero-tax year is still a filing year, and the penalties for missing an information return can dwarf any tax that was at stake.
"Accidental Americans" are dual citizens who never chose it. A child born in the United States to foreign parents, or born abroad to a US-citizen parent, is a US citizen by operation of law even if they leave as an infant and never return. They are dual citizens with full US filing obligations they frequently discover only when a foreign bank, complying with US account-reporting rules, asks for a US taxpayer identification number. Their position is legally identical to any other dual citizen's; the difference is only that they did nothing to acquire it.
The exit is a formal act with its own tax. Because the obligation attaches to US citizenship, a dual citizen who wants to stop being taxed by the United States must formally renounce it, keeping the second citizenship so as not to become stateless. Renunciation ends future US worldwide taxation, but a renouncer above certain income or net-worth thresholds is a "covered expatriate" and faces the expatriation tax, which treats most assets as sold the day before renunciation.