Yeah, I think he's absolutely the same person.
Firstly, "The Cage" portrays him at what I think is pretty unquestionably a low point - it wouldn't be fair to assume he's normally that angst-ridden.
BOYCE: Chris, you set standards for yourself no one could meet. You treat everyone on board like a human being except yourself, and now you're tired and you-
PIKE: You bet I'm tired. You bet. I'm tired of being responsible for two hundred and three lives. I'm tired of deciding which mission is too risky and which isn't, and who's going on the landing party and who doesn't, and who lives and who dies. Boy, I've had it, Phil.
The Pike we meet in "Discovery" is a couple of years removed from the mission on Rigel VII, and has been able to bounce back a bit. All the same, his character is practically defined by his faith in Starfleet's mission, and his willingness to make sacrifices in the pursuit of that mission, while also being acutely aware of the toll that takes..