I have no knowledge of VAT (because I'm US-based and don't need it).
However, customarily every invoice has a unique number - regardless of client/customer. You should never have more than 1 invoice with the same number, ever. This allows you to find the exact transaction at a later date. Having multiple invoices with identical numbers will inevitably result in a nightmare.
If you have multiple invoices, all numbered as #01.... then you'll be scouring every 01 invoice to try and find the correct one. That's not good. It may seem minor with 50 invoices, but after a couple years you may have thousands of invoices to wade through.
Again, without VAT knowledge here ... you can add a unique client identifier to invoice numbers if you'd like. I do, just to make it easier to find those "jobs" in my tracking. How you add that identifier is kind of up to you. The important aspect is that no 2 invoices ever have the same number. As far as I'm aware the only rule regarding invoice numbers is that the be unique. (If VAT is different and someone knows, please comment to let me know I'm mistaken for VAT).
For a client ID:
You could use a range of things...
#001-001 where the first three digits would be the client ID
#AAA001 where the first 3 alpha characters are the client ID.
I, personally, add the year to my invoice numbers as well as a client identifier. As in
18AAAA0001 --18 for the year AAAA for the client, followed by the actual invoice number. This allows me to typically keep invoice numbers to 4 digits since the year changes every 12 months. So I can have unique invoices numbers such as
In a list, this generally sorts the numbers by year-client-invoice which I find helpful.
Merely forward thinking -- I'd suggest all invoices have a minimum of 4 digits (if not 5 or 6) with leading zeroes. It'll merely makes things more organized once you start getting a record of many invoices.