In the country I live in it may be very handy and worthwhile if as many people as possible know their own blood type. In case of much needed blood transfusions (planned or unexpected surgery, or other emergency) relatives of the patient may be looking for people with the same bloodtype to help out. So the more people you address who know their own bloodtype, the more the chance of succes to find possible donors. And in case of networking for needed transfusions, if you are informed about blood types of other people in your environment, and them in their turn about those in THEIR environment, may save precious time and effort in the search for blood.
Also, regarding random blood donation it is good to know your blood type, sometimes there may be calls for certain blood types of which hospitals/health organisations do not have enough in 'stock' (or they are already in serious need for those), and if they for instance urgently need more of a type like B Rh- it's a good thing that possible donors know they are able to donate this particular blood type. And any other blood type is always very wanted for donations anyway.
For such things, I am very glad to know my own and my family member's blood types.
I guess that in emergencies, even if blood type is already known/ provided, it's being tested/double checked quickly to be able to do transfusions nowadays.
We both have always known our own blood types and our kids' blood types, which we learned at birth. It was good to know since their jaundice was a seldom incompatible blood type related kind (was not rhesus related, another kind) and just by knowing the blood types in our family, the type of jaundice they suffered from could be assessed immediately and the drs. could conclude this just by looking at all our blood types in the file. And we knew with the birth of our second child that him having a certain blood type would make the risk of having this same condition very high, and so on with possible subsequent siblings.
I do not think there is any harm for the parents/children to own this info for themselves apart from it being written their medical files :-).
Here a dr. always asks a patients his/her blood type. Maybe because there's no overall system where you have just one medical file, each dr. (office/department) you visit has his/her own file health file of yours with specific complaints you consulted them about. So here it's just a thing necessary for the record. And it would be silly if each and every dr. would have you (re-)tested for your blood type 'just for the file'.
If you're really curious or you would really like to know, you might ask to determine your/your child's blood type next time a blood draw is done. But I wouldn't do an unnecessary blood draw/prick just for this unless there is a good reason you think you should know (like one of the above).