I read that differential amplifiers "subtract" two input signals and amplify the difference. Here, there's no difference to speak of so why is this referred to as a differential input stage?
The fact that there is no difference doesn't mean the input stage of that circuit isn't a difference amplifier -> The fact that at any given moment you aren't doing a good deed doesn't mean that you aren't a good person.
A (negative) feedback-based amplifier works by reproducing the output in a way to null the input. You can think of it as a tracking amplifier: say that you have a 1v input signal. The output thus the other input is at 0v. So the difference amplifier outputs a signal to increase the output to (0.5v). Now, your difference is only 0.5v. The difference amplifier outputs a signal to increase the output to 0.75v, ...... In the end, the output tracks the input. We think of it as "instantenous" when in fact there is an ever so tiny delay -> stability issues with feedback amplifiers.
Is it then correct to state that the collector current at Q102 is the output of the differential input stage and is a function of input voltage + feedback and this relationship has a TANH form?
The difference amplifier's output is its current on the collectors. That current, when the input stage is biased via a CCS, is solely the function of the difference between the two inputs.
That output is linear to the voltage difference, if you allow Vbe to be constant -> mostly true for small signal swings.