Paul Smith was standing in his store on Albemarle Street in London — where he works every Saturday on the shop floor — when a fascinating customer caught his eye. “A lady came to the store as a rabbit. She was as we are now, just chatting,” says Smith, who’s been collecting lucky rabbit figurines for years. “I said, ‘Can I help you? Do you need a new tail or anything?'”

Smith’s customers, an eclectic bunch, tend to treat him like an old friend, while anonymous admirers from around the world send him inexplicable gifts — a 20-foot-long yellow pencil, carved fish, bicycles, duck and goose statues and the crinkly wrappers used to pack fresh fruit — all of which he stores in his curiosity shop of an office.

His past 56 years in business have been just as colorful.…