Yes, Finding Dory is right about this.

Having multiple hearts isn’t as odd as it might seem. Although one might be advised to keep one’s brain and one’s heart, as well as one or two other organs, separate when making important decisions, a heart and a brain are metaphorical of each other in this regard. Nervous systems can exist and function without brains, but in many animals clumps of neurons known as ganglia concentrate neural function. The same sort of electric and chemical interactions occurring across a network of neurons can have more complex functions when the neurons are grouped together. A brain is an extreme example of this. Similarly, blood vessels can have muscular tissue that contracts in a way that causes blood flow, as is the case with the arteries in human bodies. A heart is, in a way, a more extreme and complex version of that. So, worms, hagfish, and octopuses have more than one heart doing similar yet different things.

In the octopus, two hearts, called branchial hearts, pump blood through each of the two gills, and the third heard pumps blood through the rest of the body.

Time Lords, such as The Doctor, have two hearts, but the evolutionary background for this is unknown. However, it is likely that Time Lords and Old Ones have something of a history together.