Is it possible to get your animal companion killed? And if it is possible, how can a Ranger get a new animal companion? By summoning, or by taming another beast? Is it just up to the game master?
Yes, you can get your animal companion killed. Nothing on your character sheet is permanent or "safe" (DW PDF version, p. 30):
Advancement, like everything else in Dungeon World, is both prescriptive and descriptive. Prescriptive means that when a player changes their character sheet the character changes in the fiction. Descriptive means that when the character changes in the fiction the player should change the character sheet to match.
"Prescriptive change" means that having Animal Companion on your sheet gives you an animal companion, but "descriptive change" means that when you get your animal companion killed, you lose the Animal Companion on your character sheet.
Similarly, descriptive change is how you get an animal companion again: you go and befriend another animal, so that it accompanies you as your companion.
Of course your animal companion can get killed. In a really dramatic situation the GM could use a move like spend their resources or show a downside of their class or equipment (it's a living creature after all). Killing the animal companion would probably fall in the hard move category.
The best way to get an animal companion back is to ask the ranger. He knows best. Ask questions and build on the answers. This can vary from one campaign to another so there's no ultimate answer to this except the one the ranger and the group can come up with.
In the end, it all comes to what make sense in fiction. In my campaign, the animal companion of the ranger is a mechanical eagle animated by the spirit of the forest. If it would be destroyed or broken for some reason, he could probably repair it and given some time and with the approbation of spirits he could get his eagle back relatively soon.