No, when you are on the Board screen of a Next Gen project, the browser tab shows the initial Board name you chose when you started the project, and does not reflect the change for each new sprint. Some of our projects go month to month, so my browser tab says July, 2019 (since that is when I created the project / first sprint) and we are in November, 2019.
Does anyone have a screen shot of this? The OP asked about making this change in a Team-managed project, but half of the responses and screen shots are referencing Company-managed-only screens.
In my Team-managed case, I see nowhere in the board configuration page that even references a board name. But I still get an old, garbage name in my browser title that I cannot seem to edit.