My initial reaction was to wonder whether you could work out a way to bring your daughter along on the haircut outing, perhaps have her reading or chatting in a chair while she waits. However, not only patience a challenge for some five year olds, I think it is valuable to have "alone time" with each child.
If your son enjoys his special Daddy outing, your daughter will doubtless enjoy it just as much. I don't have twins, but each of my children likes going on errands with me without the others, whether we're going on a special Just For Them destination or to pick up a loaf of bread for dinner. The particular advantage of an outing for an activity is that the siblings are far removed and can’t possibly interrupt.
You mentioned that you both get a haircut and do some shopping. Consider saving some of those shopping errands for a different weekend, and take your daughter on those. If your son feels hurt that he is left home, explain that you really enjoyed spending a day with him last week and it's Katie's turn to run errands this weekend.
You can also target the shopping in each case -- include a small useful item (book, socks, whatever) that they need. Note that needing that item is important: I am not suggesting that you buy their time and love. Every child needs stuff occasionally, and having a dedicated outing that focuses on the kid's input gives you both quality together time, while helping them feel confident in their opinion and its value.
Your current haircut outings already accomplish this: your son needs to buy a service, so you take him along. Look for a similar opportunity the next time your daughter needs a pair of socks, or is buying a present for a friend’s party, or wants a particular snack -- anything her brother doesn’t really have interest in. For example, my son is the only one in the house besides me who likes a particular kind of dried fruit, and so that’s the outing that he and I tend to take together :)