Is it a common way to say goodbye? I understand it is informal, but what would be an appropriate situation/context for using it?
drücken is used in the meaning of "to hug" here. So it basically means "I'm hugging you". That wouldn't necessarily be said to someone who is present with you (you can hug them without saying so), but rather in writing or on the phone. Don't use it with anyone you wouldn't hug physically.
The phrase sounds a bit unusual to me. One would rather say "Fühl dich (von mir) gedrückt.", which is normally used to console someone but can also be used in addition to saying goodbye.