Is this function from an Elementor option of a third party Elementor addon plugin?

I can explain what happens and goes wrong. The page loads with the HTML you see. The popup button function comes with extra JavaScript code that runs when the document is ready, that attaches the popup to the button.

When you make a search, the posts are refreshed, but the popup button function does not see this happen (it is coded only to look at a full page load) so there is no popup code attached now to the buttons.

What will need to be done to fix this?

Two things. One from us, to supply you with a JavaScript code snippet to help re-attach the popup function, and then, you need the code to actually re-attach the popup. The latter must come from whoever wrote the popup function (not us, so either Elementor or the third party popup button plugin author).

The code snippet from us looks like this:

<script>(function ( $ ) {
  "use strict";
  $(document).on("sf:ajaxfinish", ".searchandfilter", function(){
    // call your script or function here
  });
}(jQuery));</script>

You would probably need to show this to whoever you ask for the code to go in place of the commented out line in my above code.