I'll work on reproducing the difference in a C program and take it up with the appropriate body.

I've arrived at:

#include <stdio.h>
#include <stdlib.h>
#include <pthread.h>
#include <mpfr.h>

void *foo( void *ptr );

int main(void) {
     pthread_t thread1;
     char *message1 = "precision in thread1:";
     int  iret1;

     if(mpfr_buildopt_tls_p()) printf("mpfr was built with TLS support
+\n");
     else printf("mpfr was NOT built with TLS support\n");

     mpfr_set_default_prec(101);
  
     iret1 = pthread_create( &thread1, NULL, foo, (void*) message1);
 
     pthread_join( thread1, NULL);

     printf("pthread_create() returned: %d\n",iret1);
     printf("Back in main: %d\n", mpfr_get_default_prec());

     return 0;
}

void *foo( void * ptr) {
     char *message;
     message = (char *) ptr;
     mpfr_set_default_prec(201);
     printf("%s %d\n", message, mpfr_get_default_prec());
     pthread_exit (NULL);
}
[download]

It exhibits the same 12.2.0-versus-13.0.l anomaly.
If that C program is compiled using gcc-12.2.0, then the foo() sub reports that default precision is 201, but if it's compiled using gcc-13.0.1 then a default precision of 53 is reported.
According to the mpfr developers, it should report 201.
I'll see if I can get the mingw-w64 developers to shed some light on what's going awry.

Cheers,
Rob