EQ and saturation are probably the main tools there.
Analog systems are usually nonlinear, both in frequency response, and on the signal level, and in some cases (such as with tube amplifier circuits), the distortion is such that it adds pleasant, “musical” overtones, as opposed to the terrible, harsh sound that tends to result from distortion in transistor based amplifiers, and digital systems.
For a subtle “all-in-one” solution aimed at a specific type of sound, you can try simulations of vinyl, tape, or tube preamps - or if you’re really serious about it, run the signal through the real thing, like an analog mixer, or a mic’ed guitar amp.
If you’re after something closer to sound design, you can use EQ, resonant filters, waveshapers, various saturation/distortion plugins etc. Some are designed to simulate specific analog hardware, with the appropriate limitations, whereas others are more “scientific,” and will generally offer more options, and more extreme processing.