Creating a generic binary in linux for all x86 machines

Question

Im trying to compile a binary of an open-source project so that our users do not have to compile it themselves.

I've noticed that some binaries created on one 32-bit ubuntu machine "A" don't work on 32-bit machine "B", with errors regarding missing .so files being reported.

However, if I compile from scratch on machine "B", then all the errors are gone.

Could there be any reason why compiling the code on the target machine makes these errors go away? I only ran "./configure" and "make" - not "make-install", So its not like I made these .so files available globally.

Could it be that the compiler detects there are .so files missing from the system library and in this case links a static library into the executable?

How does Ubuntu compile its packages so that an i386 package runs on all x86 machines?