The search overlay menu, visible in the upper right corner, uses the popup feature of Elementor builder.
Popups are call-to-action modal windows that popup and overlay the page at a specified moment or under specific triggers and conditions. Popups are designed to focus a user’s attention once they have taken an action.
To begin, navigate to the Toolbar > Edit with Elementor > Menu Pop Up/ Search
Or go to Templates > Popups > Menu Pop Up/ Search > Edit with Elementor. Control the layout, conditions, rules and styles of the popup, and then design the popup content.
How to trigger the popup on the Search icon?
1. Set the Search icon link to "Popup"
2. Add the CSS class: vamtam-popup-toggle in the "Advanced" tab of the Search icon:
Please check these guides for further information:
https://docs.elementor.com/article/472-popups
https://elementor.com/help/how-to-trigger-a-popup-on-a-link-click/