My kids absolutely loved this game!
Is anybody able to recommend similar games on GOG, suitable for kids?
I've been trawling through the games but would love to hear about games you've already tried :)

The Night of the Rabbit would be the obvious choice.

I would recommend The Tiny Bang Story and Morphopolis, they are both kid-friendly games consisting of a similar combination of puzzle and hidden-object gameplay as My Brother Rabbit.

Take a look at Pajama Sam Vol. 1 and see if they might enjoy those simple adventure titles (there are two adventure games included and 1 action-y game). They may be too pixelated now for young eyes but they've aged great otherwise.

https://www.gog.com/game/pajama_sam_vol_1

Otherwise you could wait for a sale, when the games get down to about $4. (I'd suggest that for Pajama Sam Vol. 2, which has the great Pajama Sam You Are What You Eat, but the other two games aren't as good)

Well, since My brother Rabbit is an Artifex Mundi game, I'm tempted to simply recommend the other Artifex Mundi games to you:
https://www.gog.com/games?sort=popularity&page=1&category=adventure&search=artifex%20mundi

But - some of their games might be a little too scary for (young) kids, and since you haven't specified the age of your kids (but I'm assuming they're of a younger age) I would recommend the following games instead (they should be better suited for younger children):
https://www.gog.com/game/annas_quest

https://www.gog.com/game/lilly_looking_through

https://www.gog.com/game/little_misfortune

https://www.gog.com/game/the_little_acre

https://www.gog.com/game/tsioque

Now, all these games above feature "comics-like" graphics (more or less resembling the graphics of MBR), below are two more, that feature pixel graphics:

https://www.gog.com/game/inherit_the_earth (this one is pretty old (from 1994))

https://www.gog.com/game/foxtail (this one is pretty new, actually it's still in development)

Maybe your kids will enjoy one (or all?) of these.

Another vote for Tiny Bang Story [EDIT: deleted my false claim about it not being available on GOG]

Maybe also Botanicula?

A totally different game, but very kid-friendly is the recently released A Short Hike.

It is here. SCPM even linked to it in his mention of it above. :)

I had been about to post an edit to that reply which added a joke about you perhaps not having been able to see that it was here because of German censors having, perhaps, had the game blocked in your country, having seen the word "bang" in the title and thinking that the game must contain violence. But then I thought better of posting it.
;P

Simon the Sorceror and Secret of Monkey Island.

Botanicula is amazing! Already played that one. Will check out your other recommends.. thanks :)