Why the courses menu is showing up on the search results page of the portal? The Portal is just a separate site collection (team site) inside the SPLMS web application.

That behavior comes from the default selected search option, i.e. search All Sites. When it searches all sites the search results are displayed on the root site collection in this web application. In our case the root site is the LMS organization, that's why it redirects to org's page with Courses left menu, and org breadcrumbs on top of the page. Try searching this site, and you'll get different page with results. For the All sites option it will be as is until the root site for the application is not changed.