To unlock this achievement, you must start the game as Eri 100 times. Continues do not count. However, you only need to start as the character, as the counter increases immediately when you've done so. (Meaning that you can exit the game before even the mission start text appears, if you'd so wish.)
Local/same screen 2 player game counts both characters chosen. (I reconfirmed this by looking at the stats file, "UserDefault" in "C:\Users\%UserName%\AppData\Local\mslug3", on June 18th 2022, and this is still the case.)
You might have to do an action that checks your achievement progress after starting a game with a character for the 100th time, such as rescuing a POW. The achievement might not unlock before you do that.