Former research engineer (intern) at a pulse oximiter company.
This was about 18 years ago, so the details are shaky.
Yes, this is likely the probe calibrating itself. Recall that the SpO2 is based on the ratio of two measurements. From TFA, you can see that the absorption in the IR is pretty flat, so the IR LEDs don't need to be known with much accuracy - and the IR LEDs you buy are also generally in narrow tolerances because of the telecom industry using them for fiber too. But on the Red side you have a steep slope for one of the spectra, that means that you need to have a better understanding of the wavelength of red LED you're using.
But you know where you get red LEDs? Cheaply? From traffic light manufacturers. In bulk. Apparently it's much cheaper to buy LEDs in huge quantities and pay in intern to develop a measurement device to test every single LED to determine the wavelength. (And probably sell the rest back into the traffic light supply chain.)
Even after that, the red LEDs wavelength was very dependent on temperature (your finger warms it up, for example) and even the current coming in from the probe.
I never worked on the FDA approved code, but there was a bunch of pre-processing put into the sensor to determine the red LED wavelength and make calibration changes before the signal got downstream to the main instrument.
I'm sure any pulse oximiter you buy off of Amazon doesn't go through the calibration that the FDA approved ones do.
Fun fact. As an intern when you're poor, it sounds like a good idea to do calibration studies: they put pulse oximiters on every finger and your earlobes and then have a nurse administer air to you and slowly bring your SpO2 down to 70% while watching the response of the sensors. Talk about light headed! And then they take the mask off and you pop back up to about 95% (our office was a 5000 feet, not sea level) in a minute or two. Fun and you get a $50 gift certificate.