Haven’t played it yet, but is there an actual third person gameplay mode or is it just photo mode showing DOOM Guy in all his glory?
No, under normal conditions, 3rd person camera is only available in photo mode. However, if you really want it, you can force gameplay into 3rd person via the command console, which does require using Cheat Engine to access. The game is not designed for it, so it doesn't feel great to play it this way. Also, console means PC only, obviously. On the plus side, you can also unlock the pistol. Yes, Eternal has a pistol.
Fyi, if you are interested in also fiddling with Eternal's photo mode, it can be activated via the settings and can be used in any map you've previously completed. Alsi, it's turned on ingame by pressing Ctrl + Right Alt, not just Right Alt like the hud tells you. Keep in mind it has some limitations. Photo mode is not available while climbing/swinging or during cutscenes. Glory kills are 1st person only. In 3rd person, you can freely pitch, yaw & roll it in any direction, but need to stay within a sphere with 10m radius around doomguy. The rest is easy enough to figure out.