A Catalogue Number (or Cat No.) is simply just a shortened way of describing a particular product (in this case at the pressing/distribution facility). So instead of the full record label name, artist name and album title you can give the shorter version (or essentially a reference number).
You don’t have to apply to any organisation to get a Cat No, you can make it up entirely with whatever characters you would like. Usually, for Vinyl records and CD pressings, a catalogue number will consist of a few letters to describe the record label, characters to describe the format (in this case LP or CD) and an assigned number.
So if Super Audio Mastering were to set up a record label, we might use SAM as the first part of our Cat No, and (if having a CD pressed) CD for the format. So SAMCD001 would be the full Cat No for our first release, SAMCD002 for our second release and SAMCD003 for the 3rd. If it was vinyl then it might be SAMLP001, SAMLP002, SAMCD003 and so on.
Remember that the above examples are just examples. You don’t have to follow this tradition. You can use more letters, start at whatever number you want etc. It can be anything you want it to be.