It depends on your country.
In the US, it is Bowker (https://www.myidentifiers.com), the only official ISBN issuer. You register with them and get your own imprint/publisher name (one ISBN = $125 or buy in bulk: 10 ISBNs = $295).
In the UK you can get your ISBN from Nielson (https://www.nielsenisbnstore.com), £89 per ISBN or £164 for 10 ISBN.
In Canada get your ISBN here, free: https://www.bac-lac.gc.ca/eng/services/isbn-canada/Pages/isbn-canada.aspx
In Australia from Thorpe-Bowker: https://www.myidentifiers.com.au (one ISBN = $44 or buy in bulk: 10 ISBN = $88)
In New Zealand, free: https://natlib.govt.nz/forms/isn
If your publisher assigns the ISBN (or you purchase one from a publisher), the imprint name will be the publisher’s name and not your own name. The same applies if you use the free assigned ISBN from retailers or distributors like Amazon, Barnes & Noble, IngramSpark, D2D, etc. The imprint name will be the retailer/distributor name and not your name. Most free assigned ISBNs can ONLY be used with the retailer who assigned it’ e.g. Amazon’s free ISBN can ONLY be used with Amazon.