Enabling this setting stores part of the URL in your browser session to keep the URL short.
I don't think that setting affects the shortened export URLs, so if you've used the share menu to create the shortened URL, as seen below, it should continue to work even when that setting is off.
Those links should look like something like http://localhost:5601/goto/<some hash>.
However, if you are using that setting and just copying the URL (ones that look like http://localhost:5601/qzg/app/kibana#/discover?_g=h@xxx&_a=h@xxx), I believe the link will only work in the browser that the URL was copied from, not when you share it. Also, if you clear your browser settings, the links will stop working. Those sessions are only stored in the browser session storage (hence the name "storeInSessionStorage").
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