If multiple pieces would be upgraded or downgraded simultaneously (e.g., with the Lair Aspect), is it possible to upgrade or downgrade the same piece multiple times?

If multiple pieces would be upgraded or downgraded simultaneously (e.g., with the Lair Aspect), is it possible to upgrade or downgrade the same piece multiple times?

No. Pick all of the pieces you want to replace first, then upgrade/downgrade them all at the same time.
However, it is possible to upgrade/downgrade a piece multiple times if they are replaced as part of different instructions (e.g., downgrading the same Invader with two different thresholds of Lost in the Endless Dark) or with two different actions (e.g., different players downgrading pieces in the same land with Trade Suffers at Terror Level III). Also, the special rule granted by Serene Waters explicitly allows choosing the same piece twice even if the timing is ambiguous.