I went to a school called Colby Academy when I was younger. The sign outside said "Colby Academy College Prep." I don't think that the latter part was a formal part of the institution's name, since it was referred to in general as Colby Academy, but it was certainly a prep school.
There is a film called "The Seven-Year Itch". You can also say "six-string guitar", "five-year plan", "four-wheel drive", "three-piece suit", "two-time loser", and so on. The hyphen is sometimes omitted. But in all of these cases, there is a noun.
In your sentence, "old" is not a noun, rather it's an adjective. So you'd say "seven years old" or "eight miles high" or "six feet deep", and so on.
Or six foot tall, six foot two, and others that have come up with other exercises. Not everything conforms to a rule, but we'd always use the plurals in those examples with "years old." It could be misleading in other cases, where they might be one of several accepted options, not all of which conform to the rules.