I think the key point here is to understand that an "age" is not an attribute of an individual, but a status which an individual enjoys within a particular legal jurisdiction.
The "birth date" of an individual is the calendar date on which they were born (without any importance attributed to the time of birth on that date).
If there were uncertainty in a legal context due to timezones, I expect that the default approach would be to treat the local date at their place of birth, as the date of birth.
The "age" (in years) of an individual is typically reckoned as how many anniversaries have passed since their birth date. The typical legal treatment of a Feb-29 birthday is to treat the anniversary as passing on Mar-01, in those years where the operation of the calendar means no Feb-29 occurs.
A person's current age in a particular jurisdiction, would be reckoned by considering their nominal birth date (i.e. the date as it was at the time of their birth and in the place of their birth), and then considering how many anniversaries have passed by reference to the local time of the current jurisdiction.
The implication of this is that a person can have different ages in different legal jurisdictions around the world.
And for legal purposes in general, the time or exact ordering of birth is irrelevant. Two children born on the same date, even if at different times, are the same age. No further ordering of their ages is typically recognised.
In the case of the question, Bob is older than Alice because his date of birth is Mar-07, and Alice's date of birth is Mar-08. Alice was born on a later date than Bob, by reference to the local time of the jurisdiction in which they were born.
This is the case even if there is evidence that the timings of their births were such that a certain time of day in the place where Alice was born on Mar-08, fell earlier than the time of day in the place where Bob was born on Mar-07.
As for enjoying a drink on one's birthday, Alice could not yet have celebrated her birthday in the jurisdiction, and it must be obvious to her that her 19th birthday is the following day.
That is, unless the scenario is complicated further by Alice not only being born in a different jurisdiction (which establishes the foundation for claiming she is, by some reckoning, older than Bob), but also having travelled on the relevant day from a jurisdiction where she is treated already as 19, to a jurisdiction where the calendar date has not yet turned over and she is treated as only 18.
As a final aside, it's important to recognise that exact times of birth are often not known (especially as to whether it falls exactly before or after midnight, which is when the date changes), and historically in the Western world (and still in less developed parts of the world) even the date may be subject to some considerable uncertainty as births were not routinely registered. This is resolved typically by forcing a choice to be made then making that choice binding.