I mean literally, they're the same 'person' but with millions of years difference.
Now please, stop throwing your chairs in outrage and listen... Or read... Unless you're blind.
Lets start with the first point:
Jake is a dog, not a robot. Right? Well if you remember your Blade Runner you'll remember that 1) Harrison Ford is a super badass, and 2) That at a certain level of sophistication, there is no clear distinction between a robot/android and a biological creature. Now the Bender we know was a 2nd generation robot and pretty unsophisticated. Prof. Farnsworth invented the first gen not long before and society had not fully accepted robot kind [remember prop infinity?] by the time of Futurama.
After enough generations of machines building and designing their successors, they would end up looking and functioning like biological beings. And if the 2nd gen version had extendy arms and legs, I'm sure the nth gen would have extendy everything.
Lets also look at the designation 'dog.' No other traditional animal exists in the post-apocalyptic world of Ooo. Aside from sentient worms, evil penguins, and a snail that contains the mind of the Litch, we've yet to meet anyone or anything (aside from Finn) who was a traditional earth-based life form. Yet we have Jake the 'magic' dog. It is my assertion that as human society progressed and intertwined with robots, we changed the designation of our partners to something that meant a best friend and constant companion. And what better friend has mankind ever had than the dog?
Then regarding the proclamation that Jake has 'magic' stretching powers, we must only remember Clarke's law that “ Any sufficiently advanced technology is indistinguishable from magic .”
So we've crossed the first plausibility gap. Now lets take a look at behavior.
In the episode 'Apple Thief' Jake repeatedly tells Finn about how he used to snatch purses and hock stolen goods before he learned it was bad. This behavior is [likely intentionally] reminiscent of Bender's general attitude and behavior. But when did he 'learn it was bad?' We know that all Bending units (except our Bender) are basically immortal. When they 'die' their minds are uploaded into a new body and they go on with their lives.
Well any new body would require a new set of operational programs – essentially like installing new drivers when you throw a new video card in your PC. It would be poor design if a robot mind were dropped into a new body and didn't know what internal commands to use in order to move and function. And with any new ability that their new chassis provided, there would need to be a set of moral/social guidelines on how and when to use those abilities. Therefore robot morality would evolve over time and develop along with their abilities and social freedoms/responsibilities.
At some point in a robot's near-immortal life they would 'learn' that stealing was wrong when their moral core was updated to include that definition.
Also, Jake's immaturity and fierce loyalty to his equally immature human companion mirror how Bender acted (perhaps with a more upgraded moral core – this \is\ a kids show after all). Plus Jake's eternal drive to party and inability to do anything else with much conviction are very Bender-ish as well. Such loyalty is perfectly normal for a race that was created to live with humans, and even explains why every woodland creature ignored baby Finn except Jake's parents.
And one last piece of the puzzle: the Rainicorn-Dog wars.
These wars are why Jake's romance with Lady Rainicorn is so scandalous. That these two can and do co-exist shows us that there is no inherent reason for the two species to clash, and no reason is ever given. Except one that may have missed you.
Rainicorns eat humans.
Here I assert that Rainicorns are the reason why humanity is all but extinct. As they are able to phase matter through their bodies, fly at speed, and who knows what else, they would be a nearly unstoppable enemy. Where they came from and why they speak Korean are not germane. If humanity were locked in a losing war with the Rainicorns, our faithful companions and social equals [the Dogs/Robots] would certainly fight on our side [and if it was part of his base programming that he may one day need to fight the Rainicorns, it would explain why Jake can understand Korean but never speaks it]. Jake is told by his father in a flashback that he would be hurting people his whole life. This is exactly the kind of thing a parent would cheerily say to their offspring if an eternal war was being fought.
In any conflict involving humans, robots, and an even more powerful foe, robots would certainly do the bulk of the fighting. They would likely even continue fighting on our behalf even after we had pulled our last-resort [scorched-earth] and blew a giant hole in the Earth. And for the untold ages after that, it would no longer be the Human-Rainicorn war, but the Rainicorn-Dog war. Only in the past few decades with the invention of soy-humans have the Rainicorns stopped their attacks.
(Note all the nukes in the first screen of the opening sequence)
Bender plays classical piano with perfect form.
Jake plays the viola with perfect form.
Bender can stretch his arms and legs.
Jake can stretch his arms and legs with much more precision, as well as the rest of his body (to the apparently cellular level)
Bender is fiercely loyal to his human friend.
Jake is fiercely loyal to his human friend.
Bender is a party animal, and often gets distracted from his primary task to do something more fun.
Jake is a party animal, and often gets distracted from his primary task to do something more fun.
Bender is childish and cowardly when facing his own perceived mortality (ie sick/dying).
Jake is childish and cowardly when facing his own perceived mortality (ie sick/dying).
At some point after Futurama, Humanity is living along side Robots as equals when a war with the Rainicorns begins. After the planet is heavily nuked (the so called 'mushroom wars' referenced in-series) the Robots (who now resemble and are called 'Dogs') continue to fight the Rainicorns until a stalemate is reached and the conflict is mostly forgotten. Some time after this, Adventure Time takes place.
Jake is a super-advanced bending robot, retaining only his general attitude towards life, voice and the basis of his physical abilities.