Google free std check and you will find various firms remarketing mainstream labs like Quest with web pages and Instagram ads. Maybe the same firm using multiple business names, with different artwork, etc.
The last time I checked, none would let you register under a nickname and let you never tell them your real name if you want a test in NY State. If you want a test in Connecticut or in many other states that are not NY, it was explicitly possible to choose to register under a nickname, Bondman008 or whatever.
For NY State, you need to provide a patient name and date of birth. In the past, a human lab receptionist would check registration details vs govt ID upon your arrival at the lab.
I do not recall if patient contact info -- phone and email and/or mailing address -- was optional but you could use your regular hobby ph etc, or create special hobby-health contact info to receive appointment details and results, and address if provided was never used (address is not needed for credit card verification if you follow typical hobbyist protocol discussed below). Nobody reading this will need tutoring on this step.
You specify a location (ZIP, county, etc) and the website finds testing centers, just like Yelp, I found some labs stay open to 7pm but others close early, at noon to 2pm so if you want to find a specific date and time you need to search interactively a bit. The hours available for remarketed STD tests might be more limited than general lab hours. Choose your appointment lab address/date/time, receive a code on the screen. Save the code.
A couple of years ago, you would then go do your test, and remarketer would release the results when you paid and cited your code with payment.
So you could sign up for the test, give bodily fluid samples, presumably the lab tested those samples, but you could not pay and never get results, and the remarketing firm would eat the lost $ to pay Quest. Or you could pay, and a week later, phone 1-888 results, give your code and be told results. If you hadn't paid yet, or you payment had not posted, then when you phoned in and gave your code, they would transfer you to a recording tell you how to pay, and you would not get results.
Now you need to pay before the appointment is approved and tests confirmed available. I forget exactly how the website enforced that dependency, but in the past could pay by USPS mailed money order that you buy at 711 or bodega. Recently you can pay on web by gift prepaid credit/debit card eg Vanilla when you make your appointment.
Now you have paid for your test and have an appointment for a time and place. When you go to the lab, eg Quest office, you tell them what name you used to register for the test, and they treat you like a VIP and take you ASAP (in my experience) so no waiting about.
Some lab offices are semi-automated (for whatever reason, minimizing public contact in the pandemic, cost of administrative staff, just wanting to feel high tech and modern, or wanting to capture data). You may get a kiosk to self-register when you arrive at the lab. You will give it some info for them to look up your appointment, such as date of birth and name, sort of like self-check-in for a flight at the airport. Key point here is to provide info based on the name and DOB that you provided when registering. Possibly no human will ask for your info, nor for ID if you provide info on the kiosk that matches their system.
Once you check-in via receptionist or kiosk, you get called, go to a private room, they orally confirm what you are being tested for, and you give urine and blood samples. Not sure if site-specific swab tests are available in this market.
So
if you want anonymous testing in NY, find a lab that does check-in via iPad app on the lab's iPad, and register for tests at that lab office for the tests under a hobby-health profile, and pay in advance via pre-paid gift card; remember that profile when you check in to give a sample, and also when you check for results via phone or web in case they add security questions. The firms marketing these tests might be sleazy or honest, but the labs taking and processing the samples are mainstream, like Quest.