In spectroscopy a doublet is a pair of spectral lines, usually with a very similar frequency, originating from the optical transitions between a common fundamental state and two final states that differ only in the total angular momentum value. This is very well known and is used to calibrate spectroscopic instruments in the visible region, with the sodium doublet corresponding to the wavelength of yellow light.
In particle physics, there are singlets, doublets, triplets and multiplets of particles by analogy with the groups of spectral lines related to the atomic and molecular transitions arising from changes in spin or angular momentum states. Different particles such as protons and neutrons can be described as a single particle with a different isospin state. One transits to the other by changing its isospin value, just like an electron changing its spin state in an atom will give different spectral lines associated in a multiplet.