You're being pendantic. Choose whatever symbol you want for "2" and "4". The point is that if you have symbol apples and someone gives you symbol more, than you have an agreed-upon number in total.
If we had a radically different perspective (like Borges' Funes the memorious), you can imagine how adding wholly distinct objects might seem ridiculous and derive some other wacky system of arithmetic instead.
Of course, you could alternatively derive it from set theory, but you might also end up with something fundamentally different than what the grandparent intended like presburger or skolem arithmetic.
Furthermore, you can translate “2+2=4” to any other formal system (your examples, “2+2=10 (base 4)”, “2+2=1 (mod 3)”, etc.), and it’s still true.
“2+2=4” is a universal truth, just expressible in different ways.
If you define a system where “2 + 2 = 5”, but also “a square has 5 corners”, “carbon has 5 covalent bonding positions”, etc. your system is coherent, but you actually are stating the abstract property “2 + 2 = 4” in common math, just using the symbol “5” to represent what’s commonly represented as “4”. A bit confusing, a less confusing example is common math, substituting “2” with “B” and “4” with “D”, so “B + B = D, a square has D corners, …”
If you define a system where “2 + 2 = 5, a line segment has 2 ends, a square has 4 corners, 4 < 5”, you’re objectively wrong (unless you’re taking common math and substituting more than digits)…if you extend this system you’ll find contradictions (what happens if you combine 2 parallel line segments of the same length at that length distance?), especially if you try to apply it to the real world.