The dlopen and dlclose operation is wrapped in exception handlers, _dl_catch_error.  Constructors and destructors are invoked under these handlers, which means that if the lazy binding trampoline detects an undefined symbol in an ELF constructor or destructor, this error is not a fatal error that terminates the process.  Instead, the dlopen or dlclose call fails.

# Reproducer

cat > undefined_function.c <<EOF
#include <stdlib.h>
#include <stdio.h>

void undefined_function (void);

static void __attribute__ ((constructor))
init (void)
{
  puts ("init is running.");
  if (getenv ("RUN_IN_INIT") != NULL)
    undefined_function ();
}

static void __attribute__ ((destructor))
fini (void)
{
  puts ("fini is running.");
  if (getenv ("RUN_IN_FINI") != NULL)
    undefined_function ();
}
EOF
cat > undefined-initfini.c <<EOF
#include <dlfcn.h>
#include <stddef.h>
#include <stdio.h>
#include <stdlib.h>

int
main (void)
{
  clearenv ();
  setenv ("RUN_IN_INIT", "1", 1);
  void *handle = dlopen ("./libundefined_function.so", RTLD_LAZY);
  if (handle == NULL)
    printf ("non-fatal error: %s\n", dlerror ());
  else
    printf ("unexpected dlopen success: %p\n", handle);

  clearenv ();
  handle = dlopen ("./libundefined_function.so", RTLD_LAZY);
  if (handle != NULL)
    {
      setenv ("RUN_IN_FINI", "1", 1);
      if (dlclose (handle) != 0)
        printf ("non-fatal dlclose error: %s\n", dlerror ());
    }
  else
    printf ("second dlopen failed: %s\n", dlerror ());

  return 0;
}
EOF
gcc -D_GNU_SOURCE -O2 -Wall -shared -fPIC -o libundefined_function.so -g -Wl,-z,lazy undefined_function.c
gcc -D_GNU_SOURCE -O2 -Wall -o undefined-initfini -g undefined-initfini.c -ldl
./undefined-initfini

# Output

non-fatal error: ./libundefined_function.so: undefined symbol: undefined_function
non-fatal dlclose error: ./libundefined_function.so: undefined symbol: undefined_function
./undefined-initfini: symbol lookup error: ./libundefined_function.so: undefined symbol: undefined_function

As you can see, fini is invoked twice, the second time at process termination.  At that point, there is no exception handler installed, so the lazy binding error is fatal.