A student is a classmate, schoolmate, etc. to another student. But what is a teacher to another teacher?
Couldn't it be colleague? Surely not, because a colleague could be any employee working with another.
The generally used word is in fact colleague. You may also hear phrases like 'fellow faculty member' or 'fellow teacher' or 'coworker'.
There is no specific variant of the word for educators.
You are right that classmate is more specific than colleague, and that more context can be inferred from the word classmate than from the word colleague. But I don't think the implied assertion – that there must be a similarly-specific word for teachers – is correct.
Similarly, I think you might have a hard time finding a specialized word for "fellow custodian" or "fellow cafeteria worker" or "fellow guidance counselor" or "fellow bus driver." Not every group at the school will have a word that says, essentially, "one of us."
I thought of one candidate word – faculty – but I don't think that's a precise fit. A school might have a faculty lounge and a student lounge, but it would not have a classmate lounge. In the student lounge, a student might run into a classmate, but, in the faculty lounge, a teacher would run into a... ?
Colleague; that's what I'd say.
I would say faculty mate (sometimes hyphenated or closed) fits. Here's an example from a biography of a former president of Drew University, John F. Hurst, who earlier worked at "Hedding Literary Institute, Ashland, N.Y. to teach belles lettres; where... Catherine LaMonte, whom he married afterward, was a faculty mate."
In describing the TV show "Glee," The Ft. Lauderdale SunSentinal states that this about the high school teacher in charge of the glee club: "The only positive energy he gets comes from a shy faculty mate."
A NY Daily News interview with Tom Cooley, dean of NYU's Stern School of Business with whom he "he has a healthy difference of opinion with the doom and gloomers of his profession, notably bearish facultymate Nouriel Roubini" (Roubini is also in the Stern School).
I'll admit that the term is used considerably less than colleague, but one does have the option of being a bit more specific.