It seems like you're either suspecting them of cheating or jealously noticed the usernames of people they've searched on a social platform like Instagram. Does that sound about right?
My advice is that finding where this data may or may not be stored is not the solution. The solution is having a partner you trust and engaging in good communication in the toughest of times.
If there's something that concerns you, raise this issue gently with your partner. Give them the opportunity to explain before you go snooping down a path that will cause more problems. Or maybe your concerns will sound crazy if broached aloud, in which case, maybe you're overreacting, reading into things that aren't there, and need to calm your nerves- go for a walk, meditate, listen to some music, etc...
It's likely that this data was cached and the search history or app data has since been cleared. If that's the case, it won't be long before it's populated again- people are creatures of habit. You'll likely have the chance to "snoop" in the future if the situation is dire... but it seems like you're looking for something that you don't actually want to find. That's an endless search that only turns up two things: emptiness or heartbreak.
