When does a new day start?
I live in the UTC+2 timezone, so I asked my friend who lives in the UTC-7 timezone to send me a screenshot so I could confirm that the time the "next day" starts in Mario Kart Tour is one, global moment and not dependent on some timer per player. After doing the math, our times did line up (and I found out, interestingly, that the Daily Select items are different for each person).
Additionally, I ensured that the game does switch the "hours remaining" on the hour exactly (instead of rounding up or down based on the minutes) by taking a screenshot at 21:55 and then again at 22:01. Sure enough, on the hour is when the "hours remaining" switches.
So according to my friend's time zone and my time zone (e.g. right now at 22:00 there are 9 hours remaining and I am in UTC+2), the game switches to a new day at 6:00am UTC.
I haven't stayed up to check, but it seems to be 11:00 pm for me, in PDT (UTC-7). The store resets in "14 hr", while it's currently 8:45 am, and all times and dates seem to be represented as the shown time plus some change.
If correct, this would make the rollover time 6:00 am UTC, or midnight in MDT/CST.