We haven't used the dining plan in awhile, so this is from memory and correct me if I've gotten it wrong.
One of the first things your server will ask is if anyone is one the dining plan. Just tell them you all are...on the same plan. Once you are finished dining, the server brings the check and will scan
your magicband. At that time, the appropriate number of adult and children's meal credits will be deducted from the number available. Your receipt will show the number of remaining credits available.
Think of it this way for your family:
5 night stay with 3 adults, 1 child means
A. Total of 45 TS/QS credits (3 per adult per night for 3 adults) for the adults on the dining plan to use;
B. Total of 15 TS/QS credits (3 per child per night for 1 child) for the children on the dining plan to use.
Remember, any child 10 and up is considered an adult on the dining plan. So, you're going to have at least 6 adults sharing those 45 TS/QS credits; if their child is 10 or older, that's 7 adults.
Just know that by doing this, you will end up paying out of pocket for meals if you plan on sharing with the other family the entire length of your trip, since you will get either 6 or 7 adult meals for the entire group of adults. That covers lunch/dinner for three days of your trip.
I'd take
@21stamps suggestion...each family gets their own regular dining plan (1 TS, 1QS, 1 snack per person per night). It may be cheaper in the long run when you realize the number of meals both families will end up paying out of picket once all dining credits are used. And it's impossible to make a meal out of your snack credits.
Hopefully, the other family is staying on property.