"Natural alignment" is a term of art meaning that the address (or in this case, offset) of the data is a multiple of its size. It can be used independently of the target, and for any size, though it's usually applied to powers of two. For example, on most platforms with multiple page sizes, pages must be naturally aligned; you can't construct a 1MB hugepage from an arbitrary span of 256 4kB pages, but instead have to use a region aligned to a multiple of 1MB.
In this case it would mean aligning 64-bit fields to an offset which is a multiple of eight bytes, even on i386 where only 32-bit alignment is required.