Yes, TBO is a manufacturers recommendation, and though they are required to publish one, the owner is not required by regulation to comply with it. As long as the engine can pass an inspection (compression test, ect.) it is legal to fly.
As far as renting an a/c with a "timed out" engine, in my exprerience I have found that there will be a certain stigmatism about renting one in a segment of the pilot population. I was renting a twin once with one very high time engine, and would often hear from fellow instructors who knew the plane "you're flying that POS, the right engine has xxx hours on it" (the time ranged from "at least 2500" to "over 3000"). In reality it had something like 2200-2300. Engine never quit.