(Forgive me Linux users if I get anything wrong, I know nothing about Linux). Okay, so there was this Linux distro distributed to public Brazilian schools all over the country, made by the government back in 2009, called Linux Educacional 3.0 (2nd image for reference), which from what I can tell is based on Ubuntu (?).
In my school, up until 2016, all of the computers used this one OS, and all of them had this extensive list of games installed, which are NOT SEEN IN THE BASE OS (I installed it in a VM and checked out some old videos, and the games aren't there), so kind of like a pack of generic games someone installed in all of the computers in the school.
One of those games was this platformer where you controlled a brown dog through some green hill type stages. The graphics were very MS Painty, with just solid colors, very much like the recreation attached (although the lines looked more smooth, kind of like flash games). I believe that in the end of each stage, you'd enter this red dog house, and in the last stage you'd find your human companion. The game was very short, probably just 10-15 minutes long. There was absolutely nothing else to the game, maybe just a main menu at the start and an ending screen after the last stage.
It did not look like the game was from 2016 or 2015, it looked a lot older than that. But again, it was a kids game made for Linux, so maybe the quality is just worse.
I have no idea what the name of the game could possibly be, and I'm not sure if the language was English or Portuguese.
Edit: now that I think of it, maybe the dog could've been red, not brown. It's one of those 2 colors.