The latency value is actually due to the fact that, in order to properly reproduce what's happening when recording to tape (the non-linearities and non-causality of the system). We have to impact the signal up to 1882 samples, meaning that for a sample happening at time N, this sample has an impact of what has happened at N - 1882. The dynamic behavior of the tape is mostly influenced by what's going through it, all the transient processing etc. What you are currently recording onto the tape has an impact of what you hearing on playback.
A tape machine, and all of it's components (Machine+Tape+Heads) is a non-causal system; meaning, the output signal depends on the state of the system. This also means that the same input doesn't always give the same output, because the output depends on what has been played through the system before.
A perfect example of this is a Spring Reverb. The spring will react differently to the same input, depending on how it has been excited before. Many systems are non-causals, but they are modeled as causal because it is much easier to do - same in, same out, always. The latency is, unfortunately a by-product of authentically replicating every aspect of the tape machine - the input electronics, record head, tape, repro head, and even output electronics.