I don't think Godzilla was acting out of character in GVK or GXK personally.

I know this viewpoint has grown to be a major talking point in the fandom lately, some people where happy and excited seeing him be so aggressive and others where bothered by it but personally, I don't think he was acting any different than usual honestly, let me explain :)

Why he attacked Kong:
I've seen a few people mention that they don't like that Godzilla went after Kong so aggressively, i've also seen people mentioning that it doesn't make sense that he went after Kong when he was sensing a trace of Ghidorah but I think I can ehlp explain why he was so focused on Kong.

So as the movies and lore have explained, Godzilla/Godzilla's kind and the Kong's had an immense war in the past, we kind of know how this war started due to the novel of GVK I believe, if I recall, the Kong's started the war due to Godzilla's immense power, they where afraid of him, so they attacked him and this led to the great war between the 2 species, GXK reveals that it was Skar King who truly started the war, Skar wanted to conquer the surface, he promised his people a life of peace on the surface if I recall correctly, but his plans where anything but peaceful, when Skar attempted to reach the Surface, he was intercepted by Godzilla, Skar and his entire army where pushed back and defeated by Godzilla, this is what spurred the Kongs to being so afraid of Godzilla, he had attacked and killed those they thought where innocent, not just that, but he took on an entire army of Kong's and still won, so then the Kong's attacked him in attempt to kill him.

After the war, I believe it's mentioned in the GVK novel that the the surface-dwelling Kongs who survived the war where exiled to Skull Island as their prison, so imagine Godzilla's train of thought when he senses a Kong having left the island, as far as he's concerned, a prisoner had just escaped, for all Godzilla knows, the Kongs where all in on Skar King's plan to conquer everything, so Godzilla senses one of what he thinks is Skar's minions, not only that, but an alpha Kong at that has escaped his prison, while the remnant of Ghidorah Godzilla sensed was far more of a threat than the Kong, I personally feel that Godzilla's attitude towards Kong and being relentless still makes sense.

I can imagine something similar happening if there where to have been a pair of Mutos ready to reproduce, he would have went on a relentless side-quest to deal with them, for all Godzilla knew, this Kong was going to attempt to free Skar King and the remnants of his army or just being a Kong could have been enough to spark Godzilla's rage, the Kong's where responsible for the deaths of many of Godzilla's kind and that could have led to the Mutos growing over-confident in culling Godzilla and his kind, with all of that said, I personally believe that Godzilla's relentless nature towards Kong makes sense and is not out of character.

Godzilla attacking ships:
I believe Godzilla attacking ships and jets firing at him makes perfect sense, while he has avoided ships before, the battle ships and jets in GVK where attacking him and in his direct path and way, not only that but it could be assume that he was exceptionally angry that humans where from his point of view, attempting to help break the balance by aiding Kong and the remnant of Ghidorah he sensed, but I think it's more simple, I believe that he was defending himself from ships and jets that where directly in his way, it was mentioned in KOTM that he could have attacked the undersea base if they hadn't put unarmed their cannons and opened their blast shield as well.
Also, in the 2014 novel, I believe it's mentioned that Godzilla had swiped jets out of the sky that where shooting at him, of course there's also all of the Russian and American ships and subs that he attacked in 1954 as well.
To add, Godzilla also attacked those pirates who tried to capture Na-Kika.

Scylla and Tiamat:
I've noticed that these 2 are what people bring up the most when explaining that Godzilla had drastically changed character, if we look at the context behind the 2 Kaiju and why Godzilla killed them, it actually fits his character perfectly!
Scylla had actually been attacking nuclear powerplants and was gaining power in an attempt I believe overthrow Godzilla, which isn't the first time Scylla had stepped out of place, in Godzilla dominion, Godzilla had to fight her to keep her from detonating a nuclear bomb she wanted to feed on.
As for Tiamat, if we look back on when Godzilla first met Tiamat, she instantly attacked Godzilla, she was powerful and if it was any other Kaiju, they would have died to her, we then learn in GXK that she's a destroyer, a category of Kaiju known for bringing chaos to humans and causing unbalance, we then learn in what I believe is Monarch Declassified, that Tiamat had actually preyed upon humans for centuries, revealing that she is exceptionally evil.
With all of this said, it's likely that Godzilla had fought her before in ancient times and made her flee, similar to their fight in Godzilla's lair, either way, Tiamat was not good and Godzilla knew that, I personally don't believe Godzilla is going to respect the territory of a destroyer Kaiju, he had gone to her lair in-need of the power she was hoarding and aggressively warned her to leave, I personally think this is fully in character due to her being a destroyer-class Kaiju, she then attacks him and he has no choice but to kill her, he had given her atleast 1 chance before-hand and we know that Godzilla is forgiving but only so many times.

With all of that said, I think what Adam Windgard has done with Godzilla is fully in-character! some may disagree and that's perfectly fine! i'm not here trying to tell people what they should or shouldn't think, just explaining why I think Godzilla's behavior in the last 2 movies fits! :)