If you subscribed in the Apple TV app to the Showtime channel, then you must also watch it in that app. No sign-in needed.
If you subscribed in the Showtime app, then you must also watch in the Showtime app. No sign-in needed.
If you subscribed in the Showtime app on a different Apple device, then you may need to use the “Restore purchase” function in that app. Apple ID account confirmation may be requested.
If you subscribed with Showtime directly (e.g. on the web), then you must sign in to the Showtime app with a Showtime account.
If you have Showtime as part of a TV provider package, then you must use a different app: Showtime Anytime.
Subscribe to Apple TV channels in the Apple TV app - Apple Support
Apple TV Channels FAQ: Services, pricing, availability, and everything else you need to know - Macworld
Showtime app - App Store preview - Apple
Showtime Anytime app - App Store preview - Apple
iPad 2, iOS 8
iOS 8 is pretty old (2014–2015), and not supported by the current versions of any of these apps. That may hinder service availability.