No, you need to connect to the servers to authenticate. Launching Overwatch without an internet connection just repeatedly fails to connect to Battle.net.

Is there in fact an in-game scoreboard I can bring up during the match?

No, there's is not. In fact Jeff Kaplan himself has stated in an interview:

So we we basically stopped displaying any form of scores, kills, deaths because it really wasn't telling the story of who was doing their job properly to win or lose as a team. And really, what it's all about is, "Did you win or lose as a team?" None of that other stuff really matters at the end of the day.

You have some form of knowing if you were one of the top performers of the game by the medals and upvotes you get.
You can also view your performance in the "Tab Menu" or tab-screen. See the next section.

Are people talking about a different screen when they mention the tab-screen?

The tab-screen is not the hero select screen. Here's an example of a tab-screen:

As you can see there are some of your stats there that you can check but you can't compare them against any other player. As I've explained in the previous section, this is intentional to promote team play.

Bonus

During development, Overwatch had indeed a scoreboard that has since been long removed.
The scoreboard used to look like this: