Showing content to specific BuddyPress Member Types

Your answer is on the codex page link in your question.

// Get the member type of user 5412.
$member_type = bp_get_member_type( 5412 );

So you could do something like this:

$member_type = bp_get_member_type( get_current_user_id() );
if ( $member_type == 'dog' )
   echo 'Bark';