First, to answer your questions about Purchase Sharing availability, it does not matter who makes a purchase, it is available to all, but there are some restrictions. The app must allow the purchase/subscription to be shared using Purchase Sharing. For example, in game tokens purchased by one person will not be available to all. For a little more detail:
What types of content can I share with my family? - Apple Support
I am not familiar with ABC Mouse, so I read through the information they provided on their website and it appears their subscription is also a Family subscription that includes one adult and 3 children.I think this is where the confusion is, because it is not Purchase Sharing that will be sharing the subscription, it is the ABC Mouse subscription that should be allowing you and your other child to each have an account. I am not sure if it makes a difference that your daughter started the ABC Mouse subscription, but there is some info here on the ABC Mouse website about adding profiles to the subscription.
https://support.abcmouse.com/hc/en-us/articles/360047531314-How-do-I-add-another-Child-Profile
I would look closer at the ABC Mouse app to see if your daughter is currently listed as the parent since the subscription was started on her device and what it will take for you to be the parent on the app and then create 2 profiles for each of your children. There is a Contact link on the bottom of the ABC Mouse page that I linked too, where they may be able to provide more assistance. I don't think Purchase Sharing or even the Family Plan play a role in this as it is the ABC Mouse app that should be making the subscription to you and your other child.