I read the other question to better understand what you intend on doing. The first thing you need to understand is that there are many ways cookies can be sent and without knowing any details of your hosting environment, of what you have control over, it is impossible to state, specifically, what you need to do. So, that said, here are some things you can start with.
Cookies are part of the HTTP specification. Assuming your server is Apache, you may need to either disable some Apache modules that send cookies by default, or modify their corresponding configuration files to tell them not to send cookies at all. The exact method for doing this can probably be found at the links above (but again, without knowing what platform you are on, this information could be completely wrong). Nginx is another, popular, HTTP server. If you are using nginx, then you’ll need to explore which nginx directives are possibly sending cookies and make sure they are disabled in the nginx configuration file.
PHP is able to interact with the HTTP headers and thus, can insert cookies. Normally, when serving only images, you aren’t interacting with PHP, so, although this shouldn’t affect you, I’m including it for sake of completeness.
By default, PHP might be configured to send a session cookie. To disable this feature, you’ll need to disable automatic sessions in PHP for the domain serving the images. See https://www.php.net/manual/en/configuration.changes.php for details on how to change this setting (it can be set in .htaccess or in httpd.conf or equivalent – if you’re on nginx, you are probably also using PHP-FPM and runtime configuration settings can probably be set in the corresponding php-fpm.domain.conf file, but you’ll need to look up the exact method yourself as the configuration can vary quite a bit).
If you serve JavaScript from this domain you need to make sure you are not including libraries from other domains inside your JavaScript. Likewise with CSS as these other domains may send their own cookies as part of each request.
In the end, you really should be using Cloudflare or some other CDN that can be configured more easily to not send cookies for static resources such as images. Additionally, a CDN will serve these assets from a location that is closer to your end users and thus, reduce the time required to load the images.
If you are serving images from a site on a multisite and the URL is directly to the images (ends in .jpg or similar) then WordPress is not a factor in the equation (because PHP is not triggered and thus does not send session cookies). If you look at the rewrite rules in .htaccess, you’ll notice that they state: if the file exists on the server, then serve the file directly.
Said another way, you are using WordPress to upload the images, but not to serve them. Serving them is done by Apache (under normal circumstances), so if cookies are being sent, you have to disable them in Apache, not WordPress and not PHP.