Not only can you use the same blockchain but you can use the accounts from this blockchain. Steem is the underlying technology based on Graphene which has more than enough scalability to support multiple front ends.

The Streemit or Streem live chat would to the blockchain use the same reward mechanisms as Steemit but the front end and UX would be different, and maybe additional features could be built exclusively for live functionality. But from what I know so far, there will be a sort of text based Steem chat being developed so all they would need to do is add web rtc or something similar.

You might need some additional features of course but in general if a chatroom is the equivalent of a blogpost on Steemit, and you break the text functions into comments, it could work. The issue would be what if the live Streemit becomes more popular? But I don't see why features couldn't be built into Steem itself to support live interfaces and live streams, so it's just a matter of paying the developers which itself is just a matter of upvoting the attempts. 

References:
https://en.wikipedia.org/wiki/WebRTC
https://steemit.com/steemit/@cass/steemit-chat-foss-alternative-to-slack-deployed-with-rocketchat