How do I use a plugin’s shortcode attribute value in another function?

This isn’t how PHP works. A PHP class is not persistent, so setting a property when a shortcode is output is not going to affect subsequent AJAX requests, since the class instance is recreated on each request.

If you have a shortcode that has an “outlet” property that the user can use like this:

[wp_example_search outlet="google"]

And that shortcode outputs a form that runs an AJAX request on submission, and you need the AJAX response to know what “outlet” to use, then the only way to do that is to send that information along with the AJAX request. To do this the shortcode would need to output the relevant outlet in a hidden field or attribute so that the JavaScript can retrieve it and send it along with the request.