Why is the fighter pilot angry that Akanye saved him from Mechagodzilla?

It's impossible to say for sure, and it's been years since I saw this film. But, my recollection is that he resented Akane due to her being the sole survivor of the failed mission using "maser tanks" where his brother died. She accepted blame for the mission's failure, and the whole team resented her.

His survival indebts him to her, who he regards as to blame for his brother's death. And he may not have preferred to live.

There is a theme of Japanese soldiers preferring death to defeat or failure. It's a very old idea. I would be careful about using the phrase "any soldier" too much.

You might look at the Japanese concept of "giri", which translates as "duty" or "obligation", but can also refer to gifts meant to obligate. For instance, gifts brought to neighbors and the landlord when moving to a new residence are considered "giri", both an obligation to appease for any disruption caused by the move, and hopeful of obligation from them for future favors back.

Sometimes this obligation is unwelcome, but it's common to use "giri" in stories to compel the unwilling, even say monsters. That doesn't mean these characters are always happy about feeling obliged. Which is actually true about giri in real life, where it is often grumbled about.