There are two problems:

1. You return $mock from IO::Socket::INET->new, but that's an object responsible for mocking the class, not an object pretending to be IO:Socket::INET. I'm not familiar with Test::MockObject, but it seems it doesn't provide an easy way to create the mocked object, so you probably have to go low level and create the object yourself:

       new => sub { bless {}, 'IO::Socket::INET' },
   [download]

2. The methods of IO::Socket::INET remain mocked only while the $mock object exists. Once you don't return it from the constructor, it goes out of scope at the end of mock_me and IO::Socket::INET will behave in its original way. You need to return the $mock object from there to the test so IO::Socket::INET stays mocked.

This works for me:

#!/usr/bin/perl
use warnings;
use strict;

{   package main;

    use Test::More tests => 2;
    use Test::Exception;
    use Test::MockModule;

    my ($obj, $mock) = mock_me(  # Keeping the $mock object.
        send => sub { $_[0]->{howdy} = 1 if $_[1] eq 'Howdy!' },
        recv => sub { $_[0]->{howdy} ? 'Welcome' : 'Hiya!'    },
    );

    is 'My::Module', ref $obj, 'Correct module';
    lives_ok { $obj->connect } 'Connection and handshake success';

    sub mock_me {
        my $mock = Test::MockModule->new('IO::Socket::INET');

        my %mock = (
            # Initializing the object.
            new  => sub { bless {}, 'IO::Socket::INET' },
            @_
        );

        $mock->mock($_ => $mock{$_}) for keys %mock;

        return My::Module->new, $mock  # Returning the $mock object.
    }
}

{   package My::Module;

    use Carp;
    use IO::Socket 1.18;

    sub new { my $class = shift; bless { }, $class }

    sub connect {
        my $s = shift;

        $s->{socket} = IO::Socket::INET->new(
            PeerAddr => '127.0.0.1',
            PeerPort => 8080,
            Proto    => 'tcp',
        ) or croak "Connection failed: $!";

        croak 'Invalid handshake' unless $s->{socket}->recv eq 'Hiya!'
+;
        $s->{socket}->send('Howdy!');
        croak 'ACK barf' unless $s->{socket}->recv eq 'Welcome';

        1;
    }
}
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