Assuming we're talking about the saying and not actual old dogs. Yes, people never lose the ability to learn. We used to excuse older people and their stubbornness to adapt in the past, because they were viewed as things to be discarded. We now know life doesn't stop when you get older, and know that learning should not stop, either.
Even though I feel people's personalities are pretty set from a young age, it's still possible for people to make a change for the better. For example, someone can lose weight, adopt a healthier lifestyle, choose to stop using foul language, stop smoking, and the like. It takes a lot of willpower and dedication to change, but it's possible for people at any age to take a positive step toward it.
Apparently it is possible to teach old dogs new tricks, according to scientists, that is. They've come to the conclusion that a person (or a dog) is actually very pliable and changeable. So, even though I am who I am today, five years down the road I could easily develop all new opinions and traits.
As long as humans are alive, learning never ends. The elderly can learn any new technology and adapt. At one time, old people had to learn how to drive when cars were brand new around 100 years ago. Alzheimer's patients frequently need to come up with new ways of doing things.
I find it hard to believe that "old dogs" can learn "new tricks." I think as people, we are willing to believe that we can fully change ourselves to be something different, but I believe within we are still the same. While a person may be able to mask over their true self for a short while, I believe the true person is still there, waiting to break free.