AFAIK, billing for DTV Stream is always separate, auto-pay to your card on file (much like Netflix). That said, the same was true back when the service was called AT&T TV, and before that, DTV Now, and I'm pretty sure that those customers were getting unlimited data from AT&T Internet. I think they had to cross-validate their accounts somehow (via the telephone number on file with AT&T, I think).
Virtually nothing has changed in terms of how AT&T treats DTV Stream since it was spun off and rebranded from AT&T TV. They still offer it to you to bundle during AT&T Fiber sign-up. So if I had to guess, I'd say that yes, it still qualifies you for unlimited data on AT&T Internet. (I do know that all their tiers with download speeds of 100 Mbps or greater come with unlimited data anyhow, although very few Uverse/DSL customers live close enough to the node to get 100 Mbps.)
Best thing for you to do would be to contact AT&T customer service and see what they tell you.