The problem is ELF shared libraries and normal executables don't differ at all if you look at their ELF headers. On my Linux machine(Debian 11.4) even e_type field of the ELF header is set to shared object file as ht utility reports even when the file in consideration is an ELF executable. It looked like the only easy and reliable method to differ ELF executable files from ELF shared libraries, and for some reason GCC fills out e_type field only with this value. Nevertheless, file(1) utility can accurately tell me if the input file I give to her is an executable or shared library.
The first answer to this question suggests that the file(1)'s code should look for PT_INTERP program header: distinguish shared objects from position independent executables
This sounds reasonable because all shared libraries get loaded after the executable file is loaded in first place, so they don't need to load interpreter one more time because a normal executable would already have done it.
Also I found this in file(1) source code when I was looking how magic.mgc file is compiled:
0 name elf-le
>16 leshort 0 no file type,
!:mime application/octet-stream
>16 leshort 1 relocatable,
!:mime application/x-object
>16 leshort 2 executable,
!:mime application/x-executable
>16 leshort 3 ${x?pie executable:shared object},
and I cannot understand what the last line means but it seems I can find an answer to my question if I understand this.