they used to teach the ME's electrical circuits by replacing the electrical parts with mechanical parts.
And us electrical guys convert mechanical, acoustic, and thermal systems into electrical equivalent circuits to grok them. When the only tool you have is a hammer...
As for circuit (linear system) analysis, I'd throw in the superposition principle and Thevenin and Norton equivalent circuits as well. Equipped with these we can do a lot, even if just for DC or DC-equivalent circuits.
Also very useful is remembering to exploit symmetry to simplify and understand circuits. This, for example, allows us to conclude that when two (identical) speakers are wired in series and fed from a voltage source, each speaker behaves as if it were fed from a voltage source. (For the thought experiment, drive each speaker from one half of a center-tapped (ideal) transformer, and then observe that, due to symmetry, the current in center tap is always zero.)
--Frank