Going to give a bit of an info dump here!
Monzo.me for current accounts has been available for users who previously had a prepaid card for some time now. The way this worked was by redirecting funds through a closed prepaid account. The new change is that inbound funds coming through Monzo.me now go directly to your current account. This allows us to offer the feature to those who never had a prepaid card. 
The rollout isn’t complete yet, I haven’t checked what it’s currently at but it should be available to 100% of accounts in the next week or two.
As for names, there isn’t a set in stone policy but there are a few guidelines that we follow. Your Monzo.me URL must be a recognisable form of your preferred name (that must itself be a recognisable form of your legal name with a few exceptions).
By default, the current version of the service takes your full preferred name and checks if it is available. If not, it will by default start adding numbers.
As with names on cards, we can use initials for middle names, drop middle names, add parts of names that were missing, use common shorter versions of names, or other forms but we can’t assign anything that either isn’t a form of your legal/preferred name or allow things too close to short usernames.
As an example, take an example user with the legal name of Abigail Emily Exampleton.
In the most common flow of Monzo current account signup where they haven’t indicated otherwise, this person may have the preferred name of Abigail Exampleton and be given /abigailexampleton but if they’re known by their friends as Abi, we could later change both their preferred name and Monzo.me link to Abi Exampleton and /abiexampleton.
Depending on the exact context and situation, other valid links may include:
/abigailemilyexampleton (Prepaid style, uses full legal name)
/abigaileexampleton (Initial used for middle name)
/abigailemiexampleton (Shortened middle name)
/abiemiexampleton (Shortened first and middle name)
/abieexampleton (Shortened first and initialised middle name)
/emilyexampleton (Middle name used as first)
/emiexampleton (Shortened middle name as first)
Be aware that the above list is not exhaustive and it is up to support to determine if it’s allowed. We reserve the right to say no but you can always ask! 
A very similar process is used for determining allowed names on your Monzo card when contacting support. This is especially relevant for longer names as we only have a limited amount of space on the card to print. We have some automated rules in place to attempt to shorten most names in the app (only by initialising or fully dropping, we don’t believe it’s appropriate to use other short forms without consent) but these don’t always work for everybody. 