Is there a switch in objdump or readelf which can tell if an ELF binary was built on Redhat or SUSE? I only have binary and no source code.
Is there any other way (like strings command or nm) that could I could use if objdump/readelf isn't useful.
Unlikely. You may make a guess based on the required libraries from readelf -a /path/to/binary | grep -E '(NEEDED|@GLIBC_)'
what the (highest) required glibc version is.
Maybe, in case there are enough strings left in the ELF binary, there is a hint if the used compiler is mentioned somewhere.