/<TABLE(.*?)>(.*?)<\/TABLE>//si)
          ^     ^
          |     |
           \   /
           added
[download]

The dot in a character class matches a literal dot, not any character. You'd need to say "(?:.|\n)" or use the /s modifier to get the dot to match newline also (and then don't use the character class).

You'd need to say "(?:.|\n)"

or "(?s:.)"

Since the file you are reading seems, judging from the sample, to contain blocks separated by a double newline, you could set $/ (input record separator) to "\n\n" which would give you one <TABLE> .. </TABLE> block as one record, to process that record further as a multi-line string with a m//mg.

But you could also use a flip-flop:

my $table_name;
my $sqldef;
while (<>) {
    if (my $flipflop = /<TABLE name="([^"]+)"/ .. /<\/TABLE/) {
        if ($1) {
            $table_name = $1;
            next;
        }
        elsif ($flipflop =~ /E0/) {
            print "table name: $table_name\n";
            print "sqldef: |$sqldef|\n";
            $table_name = $sqldef = '';
        }
        else {
            $sqldef .= $_;
        }
    }
}
__END__
table name: table1
sqldef: |FieldName1 VARCHAR(20)
FieldName2 INT(20)
FieldName3 BOOL
|
table name: table
sqldef: |FieldName1 VARCHAR(20)
FieldName2 INT(20)
FieldName3 BOOL
|
[download]

See ".." in perlop.

Almost the same as shmem, but I thought I'd use "</TABLE>" as the record separator. That gives you one extra record at the end, which you can easily ignore, but works if you don't have "\n\n". Or flip-flop; or slurp-n-split.

use strict;
my $line = '='x60 ."\n";
#&RS;
⃜
#&SlurpAndSplit;
sub RS {
   $/=q{</TABLE>};
   while (<DATA>){
      if (s|\s*<TABLE (.+?)>|$1|) {
         chomp;
         print $line.$_;
      }
   }
}

sub SlurpAndSplit {
   $/=undef;
   chomp(my @tables = split(/<\/TABLE>\s*/, <DATA>));
   print map { s|<TABLE (.+?)>|$1|; "$line$_" } @tables;
}

sub DotDot {
   my @wanted;
   while (<DATA>) {
      chomp;
      if (/<TABLE / .. /<\/TABLE>/ ) {
         if (/<TABLE (.+?)>/) {
            push(@wanted,$1);
         } elsif (/<\/TABLE>/) {
            print $line, map {"$_\n"} @wanted;
            @wanted = ();
         } else {
            push(@wanted,$_);
         }
      }
  }
}

__DATA__
<TABLE name="table1">
FieldName1 VARCHAR(20)
FieldName2 INT(20)
FieldName3 BOOL
</TABLE>
<TABLE name="table"2>
FieldName1 VARCHAR(20)
FieldName2 INT(20)
FieldName3 BOOL
</TABLE>
[download]

$file =~ m{<TABLE \s+ ([^>]*+)>
           (
              [^<]*+ (?: < (?!/TABLE>) [^<]*+ )*
           )
           </TABLE>}x;
[download]