One important thing to note is that Neurology is heavily driven by the subspecialties within it, and that can very much determine your patient base. Someone who specializes in Epilepsy can oftentimes have a very young patient base, especially in areas where pediatric neurologists are few and far between. On the other hand, someone who specializes in Movement Disorders (particularly Parkinson's) or Cognitive (Dementia) will have a much, much older patient base. MS, just like you're taught in med school, is most commonly diagnosed in youngish-middle aged women, and obviously the patient demographic reflects this. I'd say stroke is an "old patient" specialty, but being from the South, it's not unusual for us to have stroke patients in their 30s (Woo Stroke Belt! And Diabetes Belt! And Obesity Belt 🙄) And a general neurologist will do just that- see a little bit of everything.
This isn't to say that one particularly subspecialty or another will preclude you from ever seeing patients from a variety of age groups. I personally know of a 20 year old with FTD and have helped take care of a patient in her 90s with intractable epilepsy.
Personally, I think that's the beauty of Neurology. It encompasses so much- so many diseases and so many types of people. You'll hear us brain geeks say it over and over again, you'll never be bored as a neurologist! It certainly isn't all strokes and Alzheimer's (unless, of course, that's your cup of tea!)