In a word, context. Try it like this:

$self->log(LOGDEBUG, 'EDIT_USER: '. Dumper(
    { 
        state       => 'db_user',
        usr_id      => $self->client->param( 'usr_id' ),
        edit_usr_id => scalar( $self->client->param( 'edit_usr_id' ) )
+,
        mode        => $self->client->param( 'mode' )
    }
  )
);
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Assuming that produces the desired result, the problem is that in your first example, you are calling $self->client->param( 'edit_usr_id' ) in a list context, whereas in your second example it is called in a scalar context.

self->client->param( .. ) probably finishes something like:

sub param {
 ...
    return unless exists ...
    return ...;
}
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That is to say, it executes a bare return statement if the parameter does not exist. In a scalar context, a bare return will assign undef to the scalar (lvalue). In a list context, it will return the 'empty list' (usually denoted by ()), but when an empty list is incorporated into a larger list, it is folded to nothing. Essentially, it disappears. Eg:

Perl> sub x{ return };;

Perl> @b = ( 1,2,'a', x(), 5, 6 );;
Perl> print "@b";;
1 2 a 5 6

Perl> @b = ( 1,2,'a', scalar(  x() ), 5, 6 );;
Perl> print "@b";;
Use of uninitialized value in join or string at (eval 13) line 1
1 2 a  5 6
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In the former, sub x() is called in a list context, so the return statement returns the empty list and that get folded and disappears making no contribution to the state of @b.

In the latter, it is called in a scalar context, so it returns undef, which causes the 4th element of @b to take the value undef.