There is no level cap.

There is a level exp cap, that scales until you hit level 20:

As of a recent update, every level past level 20 now only requires 20,000 exp to reach the next rank, and this is consistent even after "prestiging". So once you hit Level 100, then reset back to level 1, but with a new prestige border icon, you will still need 20k exp to get to the next level, and so on.

New portrait borders are given every 10 levels, starting at 1, 11, 21, 31, etc.

After you hit level 100, you "prestige" and reset your level to 1, but gain a new portrait border. This happens every time you hit level 100 again.

At level 1890, or approximately your 18th "prestige" and at level 90, you will finally gain the last portrait border which appears gold and although you can continue prestiging over and over again your profile border will no longer change.

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: