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..

I've always preferred to skip "The Cage" and considered "The Menagerie" the definitive view of Pike's enterprise. It may appear to be a subtle distinction, but I think viewed through this lens the answer to your question becomes a much stronger "Yes".

When James Bond was no longer portrayed exclusively by Sean Connery it was pretty jarring. But I got used to it, and after a while I didn't really think much of it.

Jeffrey Hunter is pretty wooden in the role (as are almost all the actors in The Cage). He comes across as a flat, somewhat listless characters with issues with women. Anson Mount plays the role with a much better sense of the humanity of Pike.

I'd love to see a literal remake but there are scenes that just wouldn't work (eg. the creepy scene where the young female ensign speaks to him).

Exhibit A would be the Pike that we see in SNW: “Among the Lotus Eaters” who damn nearly killed Zac Nguyen while a blank slate, before his memories and restraint were restored.

So yes, that Pike from 5 years ago is still under there. He grew as a person.