Route Quiz shows you six Pokémon that all spawn on the same route and asks you to pick the route from a list. The data is from the original Red, Blue, Yellow, Gold, Silver, Crystal, Ruby, Sapphire, and Emerald. Vague memory of those games will not save you here. The game wants specifics.
I am not a competitive Pokémon player. I just replayed Crystal a few times as a kid. That is roughly enough to get to a reasonable score if you approach it in the right order.
Identify the region first
Before guessing a route, work out what region you are in. Each region has a handful of mons that only appear there in Gens 1 to 3. A Sentret or Hoothoot in the lineup means Johto. Zigzagoon or Poochyena means Hoenn. Nidoran or Mankey is a strong Kanto signal, since those mons appear in iconic Kanto early routes even if they exist elsewhere.
Once you have the region, your candidate list drops from over a hundred to about twenty. From there, most routes have one signature spawn that locks the answer.
Signature spawns are the actual cheat
Some mons appear on basically one route in their generation. Learn those and you will solve a lot of puzzles before reading the rest of the lineup.
Kanto. Tauros and Chansey are Safari Zone. Voltorb on a route means Route 10, near the Power Plant. Diglett means Diglett's Cave. Farfetch'd is trade only in most Gen 1 games, so seeing one usually means a special area.
Johto. Dunsparce is Dark Cave only. Stantler is Route 36 or 37. Wooper is Union Cave or the Ruins. Slugma is around Route 16 or 17 near Cinnabar.
Hoenn. Spinda is Route 113, the ash fall route. Skitty is Route 116. Numel and Spoink are Mt. Chimney. Plusle and Minun share Routes 110 and 117. Absol is basically Route 120 and nowhere else.
Route character as a tiebreaker
Every route in Gens 1 to 3 has a vibe set by the terrain. Grass routes are bug plus bird plus small mammal. Water routes are Tentacool, Magikarp, and a regional fish. Goldeen in Kanto, Qwilfish in Johto, Wailmer in Hoenn. Caves are Zubat, Geodude, Onix, plus one regional rare.
If the lineup has a clear theme, three Water types plus a Tentacool, you are looking at a sea route. If the lineup looks like a random mash of unrelated types, you are probably staring at a Safari Zone or National Park, which mix species on purpose.
The routes that exist to confuse you
Some routes have weird tables that exist mostly to surprise the player. Kanto's Route 23 has high-level versions of low-level mons plus rare Ditto. Johto's Route 47 has Spinarak, Sandshrew, and Ditto in a combination that feels like nowhere else. Hoenn's Route 119 has Tropius, Linoone, Zigzagoon, Oddish, and rare Kecleon. Very distinctive once you know it.
These weird routes show up in puzzles more often than they should, because their lineups are hard to mistake. If a lineup feels off, too many high-level mons, an out-of-place rare, put those routes near the top of your guesses.
Surf and Old Rod are mostly noise
Surfing on almost any Kanto water route gives Tentacool. Surfing on almost any Johto water route gives Tentacool plus, depending on time of day, Mantine. So if the lineup is Tentacool, Tentacool, Tentacool, and three weird ones, ignore the Tentacools and focus on the rest.
Old Rod is always Magikarp, which is not a useful clue. Good Rod and Super Rod give region-specific fish, which is useful for region ID but rarely pins a single route.
Track the ones you keep missing
Route Quiz only has five rounds per daily run, so you cannot grind it. When you miss a route, write down the lineup and the actual answer. After a month you have ten or twenty puzzles that act as a personal weak spot map. Reading that list before each daily run lifted my score more than anything else.