There are no explicit rules, but the intent seems to allow it
A ranger's companion seems to be considered an NPC (rules as intended)
Jeremy Crawford has called a ranger's animal companion an NPC on multiple occassions:
Want an NPC companion? No one needs a special class feature to have a companion. Roleplaying, skills (such as Persuasion for intelligent creatures and Animal Handling for beasts), and the adjudication of the DM are all you need.
No one has to take a subclass to have [an animal] companion, and the DM controls whether NPCs decide to stick around.
Special NPCs often get to use PC rules instead of monster rules
That would put it into the potential category of "special nonplayer characters" as outlined in the rules covering Monsters and Death:
Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws. Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.
Note that this is explicitly in the realm of DM rule here, but there is indication that NPCs making death saving throws is not only allowed but "common". Being that animal companions seem to be considered NPCs, applying that rule here seems very reasonable.
Allowing death saving throws does not seem to have many significant downsides
The only downsides that I can think of from my experience at several tables where animal companions did get death saving throws are that it occasionally adds more dice-rolling and bookkeeping potentially slowing down certain combat situations.
Allowing death saving throws is generally more fun
Generally, nobody likes to have a companion die. Thus, a rule making companion death less likely to occur is probably going to mean less stress and a bit more fun at the table. Obviously, the threat of death is there and real (just as it is for the PCs), but allowing them to be treated the same at least reduces the possibility that a freak accident or crit instantly kills a potentially treasured and dear companion with no chance of prevention.