Package versions:

binutils-2.35.2-24.el9.x86_64
gcc-11.3.1-2.1.el9.x86_64

This script produces a shared object with the expected alias:

cat > t.c <<EOF
int __attribute__ ((symver ("OPENSSL_strcasecmp.1"),
                    symver ("OPENSSL_strcasecmp@@OPENSSL_3.0.3")))
OPENSSL_strcasecmp (const char *a, const char *b)
{
  return 0;
}
EOF
cat > t.lds <<EOF
OPENSSL_3.0.1 {
  global: OPENSSL_strcasecmp;
  local: *;
};
OPENSSL_3.0.3 {
  global: OPENSSL_strcasecmp;
  local: *;
} OPENSSL_3.0.1;
EOF
gcc -Wall -O2 -shared -fpic t.c -Wl,--version-script t.lds

But replacing the final command does not:

gcc -flto -Wall -O2 -shared -fpic t.c -Wl,--version-script t.lds

/usr/bin/ld: /tmp/ccDIqhWD.ltrans0.ltrans.o: in function `OPENSSL_strcasecmp':
<artificial>:(.text+0x0): multiple definition of `OPENSSL_strcasecmp'
/usr/bin/ld: /tmp/ccDIqhWD.ltrans0.ltrans.o:(*IND*+0x0): multiple definition of `OPENSSL_strcasecmp'
collect2: error: ld returned 1 exit status

The assembler difference from LTO non-LTO mode appears to be:

 .LFE0:
 	.size	OPENSSL_strcasecmp, .-OPENSSL_strcasecmp
-	.symver	OPENSSL_strcasecmp, OPENSSL_strcasecmp.1
 	.symver	OPENSSL_strcasecmp, OPENSSL_strcasecmp@@OPENSSL_3.0.3
+	.symver	OPENSSL_strcasecmp, OPENSSL_strcasecmp.1
 	.ident	"GCC: (GNU) 11.3.1 20220421 (Red Hat 11.3.1-2)"
 	.section	.note.GNU-stack,"",@progbits

And the symbol table order is similarly transposed:

     7: 0000000000000000      0 SECTION LOCAL  DEFAULT        4 
     8: 0000000000000000      3 FUNC    GLOBAL DEFAULT        1 OPENSSL_strcasecmp
-    9: 0000000000000000      3 FUNC    GLOBAL DEFAULT        1 OPENSSL_strcasecmp@@OPENSSL_3.0.3
-   10: 0000000000000000      3 FUNC    GLOBAL DEFAULT        1 OPENSSL_strcasecmp.1
+    9: 0000000000000000      3 FUNC    GLOBAL DEFAULT        1 OPENSSL_strcasecmp.1
+   10: 0000000000000000      3 FUNC    GLOBAL DEFAULT        1 OPENSSL_strcasecmp@@OPENSSL_3.0.3

I'm not sure if these are the differences that trigger the linker failure.

Nick, does this look like a binutils issue to you?