Does 'filter' (as it is used here) have a precise mathematical definition? The only other place where I've seen this word used is the 'bloom filter', which according to Wikipedia is not an algorithm like the Kalman filter but a data structure.
In general, filters do what their names suggest -- they filter out noise or unwanted components of signals. The simple Kalman filter filters out white noise to estimate parameters that describe a system; other filters use things like the Fourier or wavelet transform to, say, select certain frequencies and remove others.
IIRC, The kalman filter has a state-space formulation, in which it is clear that it is equivalent to an LTI filter (https://en.m.wikipedia.org/wiki/Linear_filter). However, I believe updating the statistics of the system in response to updated measurements breaks the "time invariant" part.
This is precisely the classic concept of an audio filter, which is, I believe, the inspiration for the term "filter" in other contexts. Bloom filters and selfie filters don't have much in common in a technical sense, but if you squint you can see the connection.
Filter has very established definition in Signal Processing, and covers things which can do a lot of things, essentially something that can manipulate how energy or power is distributed over a set of frequencies.
A common analogy is yo imagine it as an entity that can turn a block of stone to a sculpture, by removing material.
It is a filter in the sense that it can track the system state from one (or more) noisy measurements. The filter part thus reflects the input denoising.
Does 'filter' (as it is used here) have a precise mathematical definition? The only other place where I've seen this word used is the 'bloom filter', which according to Wikipedia is not an algorithm like the Kalman filter but a data structure.