Well, not quite by “definition”, but yeah, you pretty much got it in one.
If you have a repeating decimal (no matter that there’s a million or a billion or a trillion digits before repeating), the number can be expressed as a ratio of two integers.
Human psychology is definitely involved. Humans are natural pattern matchers. It’s a great survival trait. So when we see patterns, we attach greater significance than they might otherwise warrant.
For example, if the lottery came out 1-2-3-4-5, we would see that as a special event, though 12-16-17-33-56 (or any combination of 5 numbers) is just as likely. The pattern causes us to attach significance to a random event.