That was highly enjoyable. I am not familiar with the old Japanese Godzilla movies, my only experience has been the 90's US version and the new movies that started in 2016.
I was very pleasantly surprised by how the focus was on the human side of things, and really did have teary eyes at the end. The effects were amazing and this movie definitely earned that Oscar.
AFAIK the only way for me to watch the old Godzilla movie is either buying a shit ton of dvd's/blurays from outside the Netherlands or illegally find them which is something I simply refuse to do. But my interest in them has certainly been piqued.
Have you tried the library? I don't know about the Netherlands, but my library here in Cincinnati has a lot of old movies in its collection, including many Godzilla/kaiju movies.
The film closest in spirit to
Minus One is the 1954 original, which had the same level of rich character drama and powerful anti-war commentary. The films after that quickly became more conventional disaster/monster movies and grew considerably more childish over the '60s-'70s, although there are some that have environmentalist messages. The best one is probably
Mothra vs. Godzilla -- which is really a
Mothra sequel with Godzilla as the Special Guest Villain, so one should ideally see
Mothra first. I'd also strongly recommend the 1965 non-Godzilla film
Frankenstein Conquers the World (aka
Frankenstein vs. Baragon), which is an excellent, rather dark film dealing with the aftermath of Hiroshima, and gives its title monster attributes that would later be given to Godzilla, like being mutated into a giant by radiation (rather than being naturally that size) and having exceptional regenerative powers.
The 1984 reboot, originally titled just
Gojira but now known as
The Return of Godzilla, returned to the serious political allegory of the original and is pretty good. It started a 7-film series with a loosely consistent continuity and some recurring characters, with highly variable quality but tending to stick with a more serious tone. My favorites are
Godzilla & Mothra: The Battle for Earth, Godzilla vs. Mechagodzilla 2, and the series finale
Godzilla vs. Destoroyah (it's actually meant to be "Destroyer," but that's how it's romanized), which brings back the leading lady from the original film. (All the pre-2010 reboot continuities include the '54 film in their backstory while ignoring most or all of its earlier sequels.) Some people think the second film in the series,
Godzilla vs. Biollante, is one of the best, but I'm not especially fond of it.
The Millennium series from 1999-2004 consists of six films in five different continuities. They were rushed into development after the 1998 American film bombed and plans for an American trilogy were abandoned (since Toho had planned to wait until the US trilogy was done), so they tried out three different teams and three different universes, with the intention of continuing the most popular one. Instead, the winning producers created a fourth universe for films 4-5, and when those films made mediocre profits, they cancelled the series and did one last anniversary film in yet another reality, albeit one that was a sequel to a few of the Showa-era films. These six were hit-or-miss, but the best is the third one,
Godzilla, Mothra and King Ghidorah: Giant Monsters All-Out Attack (or GMK), which is one of the most scathing anti-war allegories in the series, and also the most spiritual film in the series. It's also got pretty good character work. I also really liked the second one,
Godzilla vs. Megaguirus, thought it wasn't as strong on the allegory.
The 2016
Shin Godzilla, which should probably be easier for you to find than most of the older ones, is pretty solid, and definitely one of the more political movies, an allegory for the ineffectual bureaucratic response to the Fukushima nuclear disaster. Its commentary on Japan's standing in the world and its treatment by the superpowers has some resonance with
Minus One and perhaps with
Return of Godzilla.
I'm also quite fond of the anime series
Godzilla: Singular Point on Netflix, which is a very intelligent and mindbending science fiction series, even though it kind of feels like the kaiju are tacked onto a story that really wants to be about other things.