Dr. Watson thinks Dr. Mortimer is elderly because he carries a cane dated 1884.
Dr. Watson has long studied under Sherlock Holmes and attempted to follow his methods. When a new client arrives, he often tries to catch up with the detective and deduce things on his own. In this case, he uses the stick to try to make deductions about Dr. Mortimer, who left it behind.
“I think,” said I, following as far as I could the methods of my companion, “that Dr. Mortimer is a successful elderly medical man, well-esteemed, since those who know him give him this mark of their appreciation.” (Ch. 1)
Watson believes Mortimer to be an “old-fashioned family practitioner,” and thinks the cane was given to him in 1884 as a commemorative gift.
Holmes is pleased with Watson’s observations, but then goes on to say that Watson made several errors. The doctor was not elderly, but rather “a young fellow under thirty” who was given the cane as a gift when he left training to become the country doctor.
The incident demonstrates Watson’s attempts to keep up with Holmes, and Sherlock’s appreciation for them. Alas, Watson is often several steps behind. He made the same observations Holmes did, but drew the wrong conclusions from them. This demonstrates Holmes’s superiority, at least in the science of deduction.