I have always allowed my red wines to breathe for at least 45 minutes prior to pouring.
Often I don't drink the whole bottle, but instead cork what is left. When I go to serve on a different day, does breathing the wine provide the same benefit? Is it recommended each time I uncork the same bottle?
I don't think so. (I also assume that you are talking about dry red wines.)
You let the wine breathe to allow 1) things that need to evaporate to leave, 2) some air to be dissolved in wine. Which is why wine tastes so different before and after you allow it to breathe.
However, when you put the cork back and open the bottle again in a day or two, or even a week, the processes that ages wine for months and years does not have enough time to «unbreathe» the liquid (consume all oxygen and replenish the substances that evaporate during breathing).
Which is why I pour immediately from a previously opened bottle and let the wine stay in wineglass for a few minutes.
Some wines improve with aeration. Others, not so much. Mostly reds, but deciding which is which is a matter of experience and some information. Many younger red wines will benefit from, at least, swirling in the glass...but could also improve in a decanter left open for 30 minutes to an hour. I would guess that most people are drinking wines that are less than 5 years old. Many of these wines improve with some air. Older wines is where experience and knowledge come into play, as some need air and others will fade with time exposed.
Saving is another story. Red wine should be vacuum sealed and refrigerated, as air will continue to work on the product. ...over time, this is not necessarily a good thing.
