Yes, atmospheric water does contain small amounts of elements and minerals, and the types and amounts of minerals vary based on location. Unfortunately, since Scalene Aquaria's claim doesn't seem to address which minerals it is balancing or what 'balanced amounts' of those minerals are, so we can't be sure one way or the other whether their claim of 'balanced amounts' of minerals is actually true.
This paper details the chemical composition of rainwater collected in India. They found traces of ammonium nitrate, ammonium sulfate, calcium, magnesium, and several other elements and minerals.
This paper details an investigation into elevated lead levels being found in rainwater collection tanks in Australia, and although the lead didn't actually come from atmospheric dust as was at first believed, the fact that the word 'elevated' was used implies some amount of lead in rainwater is expected.
Finally, This paper details attempts at purifying rainwater in China, and that they were testing the removal of lead, zinc, and ammonium nitrogen from the rainwater. If they wanted a process to remove those elements from rainwater, then presumably they are common elements in atmospheric water.