Warframe has a lot of RPG elements, but they're struggling with keeping everything accessible to free-to-players without making everything such a terrible grind. Recent updates have shown they're more than struggling with it.
For the first 50 hours it's great, though!
System Shock 2 (Bioshock spiritual predecessor) is an amazing FPSRPG. The gameplay can't get any better. Amazing story, atmosphere, soundtrack, everything is amazing except graphics unless you mod, and that's because it's from 1999. IMHO it's the best FPS(RPG) ever made.
The first System Shock also has some RPG elements but SS2 is where things get full RPG.
Not exactly a shooter, but Dishonored has maybe the quality you want. RPG qualities like choosing your upgrades and playstyle, good story focus like RPGs, with choices impacting the narrative as you go. Not stat based though. Deus Ex Mankind divided. Again, not exactly a shooter, although you can definitely play it like one. Much more heavily an RPG than Dishonored.
There are lots of shooters with progression elements, which have been named, and then there are FPS that require you to play the role of the character.
Thief 1 and 2, System Shock 2, are not fully FPS but you have to think like a thief to play the first two and the second forces you into a defensive form of play (at first). I got the most out of the very flawed Far Cry 2 by trying toplay as a lone wolf mercenary in a war zone (avoiding roads, picking off enemies one at a time, hit and run tactics).
RPG/FPS hybrid are become my favorite games. Fallout 3, New Vegas, Fallout 4, System Shock 2, Deus Ex, Deus Ex: Human Revolution, Borderlands, Destiny (haven't played this one), STALKER series... a great lineage.
Destiny, from the creators of Halo, is probably what you are looking for. It gets a lot of crap talked about it, but personally I love it. Definitely give it a try to decide for yourself.
Wow lots of suggestions for some phenomenal RPGs with shooter elements (Deus Ex, Fallout). While these are incredible games it's fairly obviously not what the OP is asking for. The best suggestions are going to be fast, action FPS games with rpg systems like Borderlands and Destiny.