No, that's one of the freakiest things about things like the Busy Beaver function. There is an exact integer that BB(748) defines. You can add one to it and then it would no longer be that number anymore.
If you are refering to the idea that nothing that can't exist in the real universe "really exists", then the "Busy Beaver" portion of that idea is extraneous, as 100% of integers can't exist in the real universe, and therefore, 100% of integers are equally just "mathematical concepts". That one of them is identified by BB(748) isn't a particularly important aspect. But certainly, a very specific number is identified by that designation, though nothing in this universe is going to know what it is in any meaningful sense.
If you are refering to the idea that nothing that can't exist in the real universe "really exists", then the "Busy Beaver" portion of that idea is extraneous, as 100% of integers can't exist in the real universe, and therefore, 100% of integers are equally just "mathematical concepts". That one of them is identified by BB(748) isn't a particularly important aspect. But certainly, a very specific number is identified by that designation, though nothing in this universe is going to know what it is in any meaningful sense.