I can't find a support article directly answering your question but here is a related one mentioning using a single device for rentals (which also stream). Keep in mind that what you are doing isn't how an Apple ID is intended to be used. It is really for an individual to use so you personally would not normally be watching movies on different devices all at the same time. Yes, your family can watch them while you are such as on an Apple TV you are using, but not with your ID on another device. I am not sure if Family Sharing would help here whereby they could watch your content while using their IDs.
Rent movies from the iTunes Store - Apple Support - "But you can't stream your rental on more than one device at a time."
Family Sharing - https://support.apple.com/HT201060 - "Family Sharing makes it easy for up to six people [different Apple IDs] in your family to share each other’s iTunes, iBooks, and App Store purchases without sharing accounts. Pay for family purchases with the same credit card and approve kids’ spending right from a parent’s device. And share photos, a family calendar, and more to help keep everyone connected."
You can use Family Sharing on your iPhone, iPad, or iPod touch with iOS 8 or later, your Mac with OS X Yosemite or later and iTunes 12, or your PC with iCloud for Windows.
Set up Family Sharing - https://support.apple.com/HT201088
More information - http://www.howtogeek.com/212823/share-apps-music-and-videos-with-apple-family-sh aring-on-iphone-ipad/ (Mar. 2015)
Sharing purchased content with Family Sharing - http://support.apple.com/HT201085 - "With Family Sharing, family members get immediate access to each others’ music, movies, TV shows, books, and apps, and can download them to their own devices any time they like.*" "* Not all content and content types are available in all countries. To share purchased content, all family members must use the same iTunes Store country or region. Music, movies, TV shows, and books can be downloaded on up to 10 devices per account, five of which can be computers. Apps can be downloaded to any devices the family member owns or controls."
