I don’t see the subscription that I purchased in the iOS app on my Touchpix.com account
The Apple subscriptions you purchase within the Touchpix app are not linked to your Touchpix.com account. But they don’t have to be. Everything you do on Touchpix.com is free and does not require a subscription.
The only feature that requires a subscription is activating an event on the Touchpix app. This is the main reason you don’t see your subscription on Touchpix.com and only see it in the app.