First of all, you can see this link in your Dashboard, right under your Account ID (it's blue). When not customized, it looks something like this:
So, the "m.me" link is defined by your Facebook Page link. To change it, go to your Facebook Page and click "About" on the left panel, then find Username field in "General" section and click "Edit" to set up your new page link. You'll just have to reconnect this Fan Page to your ManyChat account then using "+ Add New Page" button to apply changes. That's it - you're awesome.
Hope this helps!