You're right, PEERS(AAPL.O) expression retrieves the Starmine set of peers. I'm not aware of any way to change the universe from which the list of peers is drawn using PEERS(AAPL.O) expression. You may want to verify this with Refinitiv Helpdesk by asking them if there's a way to retrieve say peers from Related Companies using =TR function in Excel. If indeed there's a way to do this, then you should be able to use the same expression in Eikon Data APIs.
When you customize the set of peers for AAPL.O using Peer Sets Manager, this customization should be reflected the next time you retrieve peers for this company using PEERS(AAPL.O) expression.
You could use Screener to set up your own criteria such as economic sector, geography, range of market cap or revenue for selecting peers from the universe of all public companies.
The only way I can think of to create the Analyst Coverage Overlap using Eikon Data APIs is to retrieve the list of analysts covering each company and calculate the overlap between the lists using say the value of TR.AnalystUID as unique key identifying the analyst.