Maybe not, but that doesn't mean the string you produce will be random. Studies have consistently and repeatedly shown that humans are bad at both generating and perceiving randomness. If you say to pick a number between 1 and 10, 7 is by far the most common. If you say to write down a random series of coin flips (which is the same as writing down a binary number), we consistently avoid runs of the same result (HHH/TTT/111/000) since these are perceived as being "less likely".

A unique string is not necessarily a random string, nor is a unique string necessarily a secure string. I could generate a brain wallet using the first line of text from a Shakespearean which had never been used before, but any coins I deposit on it would likely be stolen.