This super German episode begins with Cari asking Janusz
"Wie viel Kaffee trinkst du am Tag?"
I am confused with the use of "am" here. Why not "pro" tag?
Both versions mean about the same, but the subtext is a bit different. "Pro Tag" implies more of a formality, a quota or schedule, an aim than "am Tag". If you say
Ich nehme mir vor, zehntausend Schritte pro Tag zu laufen.
and in the evening you see you only did 9,800 steps so far, you kind of "have to" go on a walk to reach your quota. But if you say
Ich nehme mir vor, zehntausend Schritte am Tag zu laufen.
and you're only at 9,800 steps in the evening, it's okay. You're a little below today, maybe you'll be a little above the target number tomorrow. There's less implied obligation, less exactness.
With regard to coffee, if you said
Ich trinke vier Tassen Kaffee pro Tag.
this would mean that after only three cups you would need to speed up you caffeine intake to reach the level you somehow need ;) But with
Ich trinke (normalerweise) vier Tassen Kaffee am Tag.
the subtext implies more of a habit, an average amount. Maybe you drink a bit less one day, a bit more another day, but on a normal day it's about four cups.
There is nothing wrong with "pro Tag", but "am Tag" is also fine. The word "pro" originates in Latin and refers to division in this context, but most people would not think of their coffee consumption as an average (number of cups divided by number of days).
I disagree with the other answer that suggests that "pro Tag" would refer to an intention, it can also refer to data from the past, which obviously can't be changed.
If you want to construct a difference in meaning, "am Tag" would imply that the number of cups is similar most days. If you drink 28 cups on Sunday, but none on the other days of the week, you have 4 cups per day (pro Tag), but you would not call that "am Tag". Of course nobody expects such a pattern for coffee consumption. In fact, if you ask the question with that in mind, "am Tag" is even more appropriate. The answer may be 6 cups on work days, but fewer or none on weekends. If you ask for coffee "pro Tag" and the person is math geek going for the literal meaning, you could get an average instead.
But for practical purposes in everyday usage I would consider both can be used with the same meaning.
'Pro tag' means 'per day' or you can use "täglich" which is the same meaning.
Sie sollen pro Tag mindestens zwei Stück Obst essen.
Ich trinke pro Tag mindestens zwei Tassen Kaffee.
Kosten: 7 Euro pro Tag.
'Am tag' means roughly 'during the day, everyday or on the day'.
Ich muss am Tag der Abschlussklausur arbeiten.
Denn schließlich beginnt unser Service am Tag der Auslieferung.
Ich musste 10 Stunden am Tag arbeiten.