I'd use a similar approach to Limbic~Region, but I'd accumulate the blocks as strings rather than arrays as they consume considerably less space, which will put off the time when you need to use a multi-pass solution. Building an array of newly read IDs as you go, allows for retaining the original ordering on output.

This:

#! perl -slw
use strict;

my( $id, %accu, @order );

while( <DATA> ) {
    chomp;
    if( m[^(\S+_\S+_\S+)\s+(.+)\s*$] ) {
        $id = $1;
        unless( exists $accu{ $id } ) {
            push @order, $id;
            $accu{ $id } = $2;
        }
        else {
            $accu{ $id } .= ' ' . $2;
        }
    }
    else {
        $accu{ $id } .= ' ' . $_;
    }
}

for my $key ( @order ) {
    printf "%-10s %s\n", $key, substr( $accu{ $key }, 0, 55, '' );
    print substr( $accu{ $key }, 0, 66, '' ) while length $accu{ $key 
+};
    print '';
}

__DATA__
5390_7_9   MEWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE
EITRDFEKEN PKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
5390_8_1   MKWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE
EITRDFEKEN PKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
5390_8_2   MEWYKKIGLL ATTALALFGL GACSNYGKSA DDTVTIEYFN QKKEMTKILE
EITRDFEKEN SKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
5390_7_9   MEWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE
EITRDFEKEN PKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
LSNKDYLKRV KNGYAEKYAV NEKVYNVPFT ANAYGIYYNK DKFEELGLKV PETWDEFEQL
5390_8_1   MKWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE
5390_8_2   MEWYKKIGLL ATTALALFGL GACSNYGKSA DDTVTIEYFN QKKEMTKILE
EITRDFEKEN SKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
LSNKDYLKRV KNGYAEKYAV NEKVYNVPFT ANAYGIYYNK DKFEELGLKV PETWDEFEQL
[download]

Produces:

C:\test>junk35
5390_7_9   MEWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE
EITRDFEKEN PKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
MEWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE EITRDFEKEN
PKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED LSNKDYLKRV
KNGYAEKYAV NEKVYNVPFT ANAYGIYYNK DKFEELGLKV PETWDEFEQL

5390_8_1   MKWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE
EITRDFEKEN PKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
MKWYKKIGLL ATTGLALVGL GACSNYGKSA DGTVTIEYFN QKKEMTKTLE

5390_8_2   MEWYKKIGLL ATTALALFGL GACSNYGKSA DDTVTIEYFN QKKEMTKILE
EITRDFEKEN SKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED
MEWYKKIGLL ATTALALFGL GACSNYGKSA DDTVTIEYFN QKKEMTKILE EITRDFEKEN
SKIKVKVVNV PNAGEVLKTR VLAGDVPDVV NIYPQSIELQ EWAKAGVFED LSNKDYLKRV
KNGYAEKYAV NEKVYNVPFT ANAYGIYYNK DKFEELGLKV PETWDEFEQL
[download]