I do not think such possibility is available at new accounts since already certain time. There are likely two reasons for it:
First of all, with the growing number of Garmin Connect users, the number of users with identical names grew, so they had to make the choice between limiting the choice of names, or using unique identificators instead. Apparently they chose the second option, which s a wise decision, on my mind.
The second reason may be the permanently increasing strictness of privacy laws, especially in EU, seriously limiting the use of any data that can reveal the identity of users.
You can change your user name, but (as far as i know) not the access key in the URL, and it is also not really needed, since you can access your acount with the base address connect.garmin.com. And if you want to share the URL with someone, you can copy and paste it, or you can use one of countelss URL shortening services.