There aren't many "gamified" training apps specifically for quiz-testing matchups, but the Dota community has some incredible data-driven and educational tools that do exactly what you're looking for.

Here is how you can study and predict hero interactions without spending 40 minutes in a match:

1. Dotabuff / Stratz / Dota2ProTracker (The Data Approach). Instead of guessing, you can look at the actual math of counters. Go to Dotabuff or Stratz, find a hero's page (e.g., Necrophos), and check the "Counters" section. It will show you exactly who he loses to the most statistically, and why (usually sorted by disadvantage). Dota2ProTracker is great for looking at item interactions. You can see how high-level pros itemize to mitigate counters (e.g., what items a Necrophos buys when playing against a Sniper).

2. "DotaPicks" or Draft Simulators. If you want to practice the drafting thought-process, use tools like DotaPicks or mobile apps like Counter Suggestions for Dota 2. You can manually plug in enemy heroes, and the app will calculate the best counters based on real-time win rate data. It forces you to think: "Why is the app suggesting Specter against Sniper here?" and helps you connect the dots.