It is a good thing to teach children how to order for themselves at an early age (and with that how to behave in a restaurant). A part of this should be how to handle requests for extras like this. If the child isn't told in advance that extras are not allowed, the parent simply has to be a parent and say no. Saying no is a part of parenting and it shouldn't matter if you are Disney or your neighborhood
Walmart. The server was doing their job and I find no fault with what they did based on the description here. Like others, we always offered our DD the option of using her money when she wanted something we thought was too expensive or not necessary. She was (and still is) a very adventurous eater and would look for the strangest thing on the menu to order and so we had to reign her in at times. There were times when a server was unsure whether her order was OK with us and looked our way. This happened twice in a meal in Maine when DD was about 4. At the start of the meal, DH & I were asked if we wanted a drink from the bar and we declined. As the waitress was walking away, DD said "excuse me, I'd like a Shirley Temple". After dinner, we were asked if we wanted coffee or tea and again as the waitress walked away DD chimed in with "excuse me, I'd like a decaf Cappuccino". She knew that she was allowed both but the requests were quite unusual for a child her age and so the waitress wanted to make sure these were OK. I wouldn't have expected a server to ask if a souvenir cup was OK if that's what came with the beverage and was listed as such on the menu.
OP, as others suggested, use this as a teaching moment for your kids. Let them know that when extras are offered, they should always ask you first unless they are allowed (and then expected) to pay for things themselves. Your efforts now will serve them well as they grow up.