IIRC, support for the new Sierra-OS document tabs was added at v7.0.0. As v6.1.0 was released 2 Oct 2014 and Sierra on 20 Sep 2016, the earliest v6.x release that could have supported this would be v7.0.0 as the last v6 release was v6.6.5 on 14 Sep 2016.
If you are still able to upgrade for free, I’d strongly recommend jumping to the current v7 as the big UI change was from v5 to v6, necessitated in part by moving code off some very old frameworks and onto Apple’s current ones. By comparison, v7 ‘looks’ like v6 but with improvements - of course!
As posted here, I’ve made some updates to aTbref7’s notes on document and in-document tab bars.