WP Extended Search does not work with REST by default since version 2.0.2 This is to achieve maximum compatibility with WordPress and other plugin/themes.
To enable the support, pass wpessid in REST request with ID (for specific search setting) or without ID (for global search setting).
e.g. /wp-json/wp/v2/posts?search=technology&wpessid
or /wp-json/wp/v2/posts?search=technology&wpessid=10
PS:- I am the author of this plugin. To keep things organized I suggest do not post plugin/theme support questions here. Use relevant support forum to ask plugin/theme related question.