Then the teacher is teaching something that someone more knowledgeable in the subject will later have to unteach. I'm focusing on functional equivalences because that's how math works as practiced by mathematicians. The functional equivalences are the point, and you may not notice it, but you're also relying on those equivalences, which is why you can write "6/2" in the first place. Integers are already equivalence classes of pairs of natural numbers (which is why 2-3=3-4). Rationals are equivalence classes of pairs of integers (which is why 3/1 = 6/2). If you actually try to define any of this stuff in a coherent way, you're immediately forced to deal with equivalence as a central idea.

6/2 is a whole number. 6/2 = 3. 3 is a whole number. They are equal. Usually, they are the exact same mathematical object. It is not merely that they share properties. They are literally definitionally the exact same thing (the same set in ZFC). "n is a whole number" is a proposition. It is true for n=6/2.

If a teacher is teaching that 6/2 is not an integer, unless they are in the middle of constructing the rationals and need to make a distinction between integers and equivalence classes of pairs of integers, then they are wrong. The very first thing you do after you're forced to make that distinction is you make it go away. They shouldn't be teaching the student to hyperfocus on a specific notation or format. That's a bad lesson to teach, and is something a real teacher will need to fix later. Actual mathematics professors are happy to let you write "let <christmas tree>∈ℝ". An intro proofs professor will definitely put something like "-3.999..., -6/2, -12/6, -5/5, 0, 5/5, 12/6, 6/2, 3.999..." on a number line to illustrate the point that these are just different ways to write the same thing. Fluidity in switching through and following different notations without getting distracted is a centrally important mathematical skill.