Is what you are really saying:
Twin 1 feels she should pick Twin 2 because Twin 2 is less confident and might be upset.
You also would feel better if you knew Twin 2 had Twin 1 to look after her.
Twin 1 really wants to pick a friend to have fun with as well.
It seems as if this is really about Twin 2's anxiety, and you may not have done either of them any favours by using Twin 1 as the main strategy for dealing with it.
Sadly, it's not unusual to have issues with a more outgoing, popular twin and a shyer, less social twin. It's very easy for the quieter one to rely on the confident one and not build up their own social strategies. It's really not helpful in the long term.
It sounds as though you haven't been able to push through your own anxieties and find strategies that work for Twin 2 to build independence. This could be a good chance to find some resilience in yourself to start telling her she can do this, rather than assume she can't. As others have said, there will be many children who are anxious or who get upset, and teachers will be very used to dealing with it.
if it's any comfort, over a lifetime, twins often develop very differently to their childhood patterns. I've seen a previously confident twin struggle a bit at uni while the quieter one thrived and had a huge circle of friends.