Action RPGs
Here are what I think are Action Equvalent to Turn-Based series.
Dragon Quest/Ys – both respectively defined their own styles of JRPG.
Final Fantasy/Tales or Kingdom Hearts & Mana – Tales and FF have many parallels, whereas Kingdom Hearts is massively popular in the west like FF
Pokemon/??? – There seems to be no clear action equivalent to Pokemon, the current closest candidate is Spectrobes.
SMT/Dark Souls? Or IgaVania? – Both have dark themes and dungeon crawling, not to mention notorious difficulty for DS and Monsters from various mythos and biblical references for Castlevania.
Persona/Multiple – No clear counterpart, however there are 3: TWEWY for style, Yakuza, for gravitas, and Tokyo Xanadu for gameplay. There’s also Dusk Diver which lacks some sim elements and Zenless Zone Zero.
Fire Emblem/??? – There is no action equivalent that I know of, unless you count Fire Emblem warriors
Phantasy Star Classic/Star Ocean – Both series have Sci-fi elements and involve space travel, with each entry taking place a long time after the other.
Xenogears/Vagrant Story – Playstation Exclusives, Deep story. Sparked 2 successor series: Xenosaga & Xenoblade, FF Tactics & FF XII.
Legend of Heroes Trials/Yakuza, .hack or Ys 8 onwards- Yakuza is the most likely candidate, since both have stories that spans multiple games, many intertwining plots and twists. .hack has worldbuilding for days & a pretty good story and cast. However Ys series seems to be going more story focused since 8.