My question is - what's the nature of characteristic functions, equations and so on? Am I right in understanding that this is just the general term for naming "ways" to find some invariants of some object? Or is there some other meaning?
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There's no particular reason that mathematicians decide or don't decide to name a property of something a "characteristic ____". The definition of a "characteristic ____" should be an important definition, I guess, but then again any definition really ought to be important, otherwise it's questionable whether we should be making it. But my point is simply that "characteristic" is just a nice word that might be evocative of a sense you want to attach to the definition in question, and there's no special explicit meaning that ties together its various occurrences (characteristic function, characteristic polynomial, characteristic of a field, etc.).
You're right that several terms use this adjective, for example characteristic function, characteristic equation, characteristic polynomial and characteristic (in field theory). This is the case (at least) in English, French, German and Russian (according to Wikipedia).
If you trace the history of these terms, you'll probably find that the same adjective was carried around out of analogy. These terms were then borrowed to other languages, which explains the cross-linguistic prevalence of this phenomenon.
However, there's no formal meaning of "characteristic". There surely is a connection between different concepts termed "characteristic", but it is only one of analogy.
As an example, in probability theory we find "characteristic functions", but we also find their relatives "moment-generating functions", which are almost the same. A similar example furnish the Schläfli symbols of regular polytopes and the class field number from algebraic number theory, which are also a kind of "characteristic". So not all "characteristic" concepts are actually named "characteristic".
From the other direction, certainly characteristic polynomials and characteristic equations are related; but it's not clear how they relate to characteristic functions and field characteristic, or how the latter relate to one another. You can surely draw analogies but it's nothing more the exegesis.