In your first attempt
jQuery('#form1 select[name="select_name"]').append('#form2');
will attempt to append #form2
to #form1 select[name="select_name"]
Thinking about it... how can you append a form to a select element?
If your intention is to append the select options from form1 to form2 then you just have to switch around those statements.
jQuery('#form2').append('#form1 select[name="select_name"]');
This will append the select
options from form1
to form2