Is Earth unique in its fairly clear atmosphere?

No, the clarity of the Earth's atmosphere cannot be considered unique. We don't have to speculate about exoplanets.

You could argue the answer is no, because both the Moon and Mercury have (very, very) thin atmospheres, and these are obviously "clear".

If you regard that argument as tricksy, then we can turn to Mars. Yes Mars has occasional dust storms. In normal conditions, the optical depth of the Martian atmosphere is usually somewhere between 0.5 and 1 per airmass. (Petrova et al. 2012; Lemmon et al. 2014). Most of this extinction is caused by dust and is nearly wavelength independent. i.e. between 60% and 37% of light would travel through it's atmosphere from outside. This compares with typical extinctions of about 0.2-0.4 magnitudes of visual extinction per airmass on Earth (0.1 mag at the best astronomical sites in the world), corresponding to 80% to 69% of light passing through the Earth's atmosphere from outside (to sea level). Most of this extinction is due to dust, though there is some absorption by water and other aerosols).

Thus, though Mars is dustier than Earth on average, it is not outrageously so. It would be stretching the use of the word unique to say that the clarity of the Earth's atmosphere was "unique".