Shouldn't Poppy be a Fighter/Tank?
Nope.
Fighter is tanky DPS
Tank is tanky with CC and utility.
Poppy is not CC utility heavy, she's only tanky in the tanky DPS sense. Which is a Fighter.
Also - assassin generally means dive carry, 1-shot them. Poppy can dive a carry and 1-shot them, so there's that.
Generally speaking fighter/assassins tend to have a ton of free defense so they can build pure offense as well. And when it comes to free defense so you never need to build defense ever - it doesn't get any better than invulnerability and immunity to CC. And that's on top of Poppy's passive and armor buff.
If you meant "her aesthetic with using heavy armor a shield and a hammer makes it seem weird she's a fighter assassin", then yeah - Poppy is an awkwardly designed champion full of a lot of unusual design choices.