in common math/geometry, all natural numbers altogether I think will have density 0..

The asymptotic density of a subset of the natural numbers, is defined as the limit of (number of elements of the set which are less than N)/N , as N goes to infinity.

Ok, by this definition I believe if you consider set of natural number without say n^2 numbers, then density will be 1, or "almost all", while class we ignored and similar classes are quite large and important.

Yes, the asymptotic density of “natural numbers which are not perfect squares” is 1. I.e. “almost all natural numbers are not perfect squares”. And similarly for many other important classes. Almost all natural numbers are not prime. Etc. etc.