@Zachary Early If you plan to do this and you want to be fully legitimate, yes its good to register with your state's department of state as a single member business LLC (Limited Liability Corporation), and with the IRS as a "Disregarded Entity / Single Member LLC" and also register with your state's department of revenue. Those three things will get you an Federal EIN and a State Sales Tax License.
If you plan to go full speed ahead and are in it for the long term, its good to do that.
The total cost is "Free" for the Federal EIN and ~ $150 to $250 for the State Dept of State registration.
You need to check with your tax guy about the exact forms, but thats the general process.
You can also operate as a sole proprietor and register a fictitious name, but that won't get you the benefits of an LLC or a FEIN.
If you are a kid, just do the sole proprietor route with a fictitious name registration because the cost is almost nothing. My state charges $80 for a fictitious name registration. One of my businesses is a sole proprietorship and its fast and easy to do that.
If you want to register to use Apple's Developer program and sell your game in the app store, you also need to get a Dun and Bradstreet DNB #.
If you really plan to do this developer thing as your main form of income or as a permanent side gig, I recommend you do an LLC and file with your State's Dept of State. Then as you interact with other businesses and partner with companies such as video game distribution companies or tools companies, its so much faster and easier because you are an LLC and all that info is instantly obtainable during the various application processes.
Your income tax guy can give you the exact details as applicable to your state, and can tell you the exact forms and processes.
If you are serious about what you are doing, I recommend you go full legit.
Now if you don't register, your tax guy can still handle it as a sole proprietorship without a fictions name, so from what I understand going "full business" is not a legal requirement, because your tax guy can just handle any income as a "Sole Proprietorship without Fictitious Name Registration". But you don't get all the other benefits that come with LLC and State department registration if you stay "low key".