First let me clarify a bit about how the API and how a system like cPanel/WHM work.
The DO API - This tool provides you access to the functions you can find in the DigitalOcean control panel, namely the ability to create and destroy droplets and volumes, create snapshots, gather basic information about your droplets, do reboots, power ups and power downs of your individual droplets and manage DNS records if your domain is pointed to the DO nameservers.
cPanel/WHM - This tool provides a web interface to manage a single server (it is possible to cluster multiple cPanel/WHM servers and do more but we'll stick to basic functionality). It allows you to manage several services on that server that are commonly used to deploy web applications including:
- Apache or Nginx (Web Server)
- PHP Programming Language
- MySQL/MariaDB database server
- BIND DNS Server
- Mail server
Through cPanel you can create virtualhosts which are individual sites with their own folder, mail and DNS records on your server allowing you to provide end users access to this without getting access to the services of other users but all these sites are still run by the same web server process. cPanel will also include 3rd party tools like PHPMyAdmin which provides a web based interface to manage your databases.
To your main question: "Can I build my own cPanel". The answer is absolutely but it is almost certainly not worth the effort. Despite seeming like a fairly straightforward web application, a full stack web hosting management tool like cPanel is vastly complex both to create initially and possibly even more importantly, keep regularly updated and secure.
I have built 3 complete web hosting control panels in the past for previous companies I've worked for who wanted custom solutions for their hosting clients or for some other reason did not want to use cPanel/Plesk or one of the other commercial panels. In each case this required hundreds of hours of development, constant work on security updates and refinements going on for years afterwords and cost a LOT more than cPanel licenses would have.
There are open source alternatives out there like zPanel that are similar to cPanel and completely free. They cover most of the bases but often (not speaking specifically about zPanel as I've seen this with other panels over the years) the place where they fall down is in quickly responding to security vulnerabilities and making security updates available. Being community driven, without full time engineers means that these areas can sometimes not get the attention they require and could leave you vulnerable.
If you're interested in the inner workings of something like cPanel/WHM I would strongly recommend following some of the tutorials here to pick up the skills to manage all the services you want to run manually from the command line and configuration files first. That knowledge would be absolutely vital if you wanted to automate those processes and you will likely find that managing things that way makes a lot more sense unless you have a specific need for a shared hosting environment that end users will have access to. If you still want to create your own panel at that point you'll be armed with the knowledge of how to manage these services and can start automating it and creating interfaces. There is no one-size-fits-all API underneath a tool like cPanel, it's all command line functions and plaintext configuration files.
The TLDR here is: cPanel/WHM has a cost involved not just for the software but because they continue to support it. Your money covers the cost of managing security updates, providing end user support and keeping the panel up to date. There are free alternatives out there that provide the functionality though often without someone standing behind the tool providing timely security updates. The services that make up a Linux web stack were designed to be managed from the command line and security updates have less headaches when you use official packages so start there and decide if you want to build a panel.