I am reading a logic book in my free time and usually the inference rule of universal generalization is motivated by real-life examples: Imagine having the statement that all people with brown hair are tired in the morning and that all people who are tired in the morning like coffee. Then one can prove that everyone with brown hair likes coffee: Let x denote a person that has brown hair, then x is tired in the morning and thus x likes coffee. Since we did not assume anything about the object (in this case person) x, it seems reasonable that for every object (person) x the reasoning should hold. If it did not, that would mean that there would have to exist an object y that does not satisfy this property, but then we could just do the same proof for this very y.
Now, when moving on to abstract objects (such as natural numbers for simplicity), this is more difficult to me. I suspect this is due to the abstraction of mathematical objects (what should they be?). When proving something about all natural numbers, it is again common to let n denote any object that satisfies being an element of the natural numbers and then deduce the statement. Since we assume the inference rule of universal generalization, this then holds for all natural numbers.
Question: However, why is it reasonable to assume that universal generalization holds for natural numbers (or more generally abstract objects)? Are there any good viewpoints on this? I assume that the "obvious" one is to just follow the real life example, but I am curious if there are any other views on this, as I seem to struggle with finding better ones quite a bit.