I'm not sure whether these are supported with hosted feature service cases yet.
[let me admit first that I didn't try it myself, but i know the basic concept and wanted to pass along in case it helps and if you haven't tried it yet]
I think you should be able to achieve what you are asking for using Attribute Rules in an enterprise feature service.
Here is the help link for attribute rules: Introduction to attribute rules—Geodatabases | Documentation
Attribute Rules use Arcade expression which has a function called NextSequenceNumber() - that might help you with generating sequential numbers.
.... in addition, using other spatial and non-spatial arcade function, you can achieve things mentioned in earlier posts such as:
- prefix an auto increment number with some hard-coded string (example)
- prefix the number with the polygon a point belongs to.
[this is i know for sure ]
if your data stored in an enterprise database, of course then database supports numeric field with auto-increment values - which is similar to ObjectID. This is outside of ArcGIS though.