I have seen this on a few mobile sites - They place a small button/icon to one side of the mobile menu that us a link to the search facility or it toggles the menu into a search box.
I could be configured in the template admin, pick an icon from the font awesome selection + select a Joomla menu option and assign to the button.
It could be used for anything really not just the Search Facility eg Login
Are you sure your are posting in product assigned forum?: No
What is your CMS version?: Joomla 3.x
MySQL Version: N/A
PHP version: N/A
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Have you edited any files? If yes tell us which ones.: No
Is this personal or client website?: Personal