Prevent sitecore from creating the the sitename#lang cookie

As far as I know, {sitename}#lang cookie is set by the LanguageResolver processor.

There is a code

args.SitecoreContext.Language = lang;

And in the background, setter of the Language property of the context calls

  string cookieKey = site.GetCookieKey("lang");
  WebUtil.SetCookieValue(cookieKey, language.Name, expiryDate ?? DateTime.MinValue);

I haven't found any option to stop Sitecore from creating that cookie.

The only option would be to add a custom processor to httpRequestBegin processor after Sitecore.Pipelines.HttpRequest.LanguageResolver processor and in that processor remove the cookie from the response.

I'm not sure if it's the same processor but you might try the follow which removes the site cookie.

<?xml version="1.0" encoding="utf-8" ?>
<configuration xmlns:patch="http://www.sitecore.net/xmlconfig/">
  <sitecore>
    <pipelines>
      <httpRequestProcessed>
        <processor type="Sitecore.XA.Foundation.Multisite.Pipelines.HttpRequest.StoreSiteNameInCookie, Sitecore.XA.Foundation.Multisite">
          <patch:delete />
        </processor>
      </httpRequestProcessed>
    </pipelines>
  </sitecore>
</configuration>