Look at the URL in the error:
http://solucase.com/wp-json/wp/v2/posts?categories=%27cat_id%27
%27 is the URL encoded character for ', so you can see what’s happening is that in the URL you are requesting with AJAX cat_id is not being replaced with the variable value.
This is because the way you have used quotes when defining the URL is incorrect:
"http://solucase.com/wp-json/wp/v2/posts?categories="cat_id""
Firstly, you have opened the string with double quotes, ". but are attempting to close them with single quotes, '. You can’t close double quotes with single quotes, so the quotes are just becoming part of the string, same with the cat_id variable name.
Secondly, after attempting to close the quotes you are not properly concatenating the variable cat_id onto the string. To concatenate strings in JS you need to use the + character.
So the url argument in your AJAX call should be:
url: "http://solucase.com/wp-json/wp/v2/posts?categories=" + cat_id
Or
url: 'http://solucase.com/wp-json/wp/v2/posts?categories=" + cat_id
Either will work.
Also note that you are not adding anything after the category ID, so you don”t need to open and close the quotes again.