Same but for vterm

See the built-in buffer-face-mode.

M-x set-background-color works for me.

Probably there's a way to automatically activate it when entering into Eat.

it is possible! add a lambda to eat-mode-hook which calls face-remap-add-relative for default and, optionally, fringe faces. here's an example taken from my own dotfiles:

(add-hook
  'eat-mode-hook
  (lambda ()
    (face-remap-add-relative
     'default
     :foreground "#ffffff"
     :background "#000000")
    (face-remap-add-relative
      'fringe
     :foreground "#ffffff"
     :background "#000000")))

and here's what it looks like in practice: https://imgur.com/a/9gPCio5