A zig-zagged example: God Eater copies the basics of the "Hunting RPG" formula from Monster Hunter (hunt a big thing, gather its bits, make new stuff, player power is determined purely by gear, you play in a party of up to four people). It did, however, introduce a number of new features (including some that certain MH fans had wanted to see for years when it first released): the hunts are much faster (both in the pace of play itself as well as how quickly a single hunt can be concluded), you can carry both a melee and ranged weapon (and there's interplay between the two), you can fill out the party with NPCs instead of just one "Palico" if you wish to play solo, the aesthetic is significantly different and the monsters themselves are VERY different, and the series has prided itself on having an actual, continuous narrative - something which Monster Hunter didn't really have when GE1 released.
God Eater Offshot is more or less a blatant copy of Dead or Alive Xtreme 3. The major difference is Offshot has male characters, but the game itself has very limited features compared to DOAX3.
The initial releases of the first two numbered entries were not released outside of Asia. Given the improvements in the Updated Rereleases, however, the only real complaint about this is how long it took said releases to get English translations and exports. Similarly, Namco has never shown any interest in exporting the mobile titles (two of which have, as of 2019, concluded service) or most of the ancillary media. One of the things stirring interest in GE3 is that it will be the first initial-release numbered title to come out in English.
Meanwhile, the English release of Burst on the PSP had a different example concerning its DLC. There were certainly hopes and demands for it amongst the players, but D3Publisher bluntly stated on their Facebook page that they would not bring over any of the DLC due to localization issues (i.e. the third-party promotional/collaboration items). Happily, later releases have avoided this issue, thanks in part to Namco keeping things in-house.
The Other Darrin: For the dub versions of God Eater Resurrection and God Eater 2 Rage Burst, several of the characters from original Gods Eater Burst localization are voiced by different actors.
Sequel Gap: From a localization perspective, at least. The wait in English-speaking territories for God Eater 2 was, in a word, excruciating. The initial version released in November 2013... and, one Updated Re-release later, the Anglosphere finally got Rage Burst on August 30th, 2016, almost three years after GE2 first released (and a good solid half decade after D3Publisher's release of GE Burst).
Urban Legend of Zelda: In God Eater Burst, it was rumored that Kanon Daiba had the "God of Rare Drops" skill (which increases the chance of getting rare materials after a mission) as a hidden bonus skill (as an Gameplay and Story Integration joke about her luck). Even though it's been proven false, some players will still bring her along for luck when they're hunting for that last item they need for an upgrade.
For the Anime:
Fan Nickname: "Budget Eater" due to the high production values ufotable is putting into it, as well as the infamous episode delays detailed below. Alternately, there's also "Limited Budget Works," considering this anime started airing only weeks after ufotable's Fate/stay night [Unlimited Blade Works] which itself had the nickname of "Unlimited Budget Works" due to the incredible production values put into it, the joke being that UBW ate all of ufotable's budget and God Eater has to make do with the scraps left behind.
Troubled Production: The anime for God Eater started running into problems even before it began airing, with the first episode having its initial selective screening on June 28 in Akihabara cancelled and then having its official air date bumped back a week. The most eyebrow-raising thing of this whole incident is that ufotable is making this show, and this is the first time they've run across production problems so bad they had to announce delays twice. On the first episode, no less, which is unheard of in the anime industry. It is generally thought that the budget for Fate/stay night [Unlimited Blade Works] caused the series to have significantly lower money available for it, resulting in the project being heavily hurt.
It happened again with the fourth episode getting delayed a week and having to replace it with a special episode. Though in this case, it doesn't seem like it was ufotable's fault; the episode delay only happened on one TV station in Japan, and it aired on schedule in the rest of the country...