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 "Page Info" 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 refresh permissions by clicking the button in 'Settings' => 'General' in your ManyChat account to apply changes. That's it - you're awesome.
Hope this helps!