Generally, no. There's a group called EPSG that standardizes these. (You can look them up on epsg.io.)
However, it is possible for Esri or others to define their own spatial reference systems and assign their own numbers, too.
For example, Google defined a coordinate system and gave it an ID of 600613. (The digits resemble the letters in the word "GOOGLE".) Later, this was adoped by Esri and they gave it the ID of 102100. It eventually became an EPSG standard with the ID of 3857. Older versions of ArcGIS Server would only recognize 102100, but once the EPSG assigned it an ID of their own, newer versions would recognize 3857 as well as the deprecated 102100.