Some brandies have caramel colour and sugar added to simulate the appearance of barrel aging.
My question is: Does real barrel aging add sugar to liquors, and if so, how much? Is the amount proportional to the age?
Barrel aging does add sugar to the liquid (wine, whisky, etc.) that is aged in it. In the toasting process of a barrel, the sugar in the wood caramelizes (cellulose is a sugar polymer and breaks into sugar during the charring process). The liquid has contact with this "caramel", and due to additive maturation, some of it goes into the liquid.
The amount of sugar heavily depends on the toasting grade, time of aging and time a barrel was already used, but in general, it is very low. Sugar concentration analyses of whisky showed an amount ranging from 150 mg/l to 400 mg/l which is too low for human perception. If the barrel had some previous contents (sherry for example), the sugar concentration can be much higher.
See for details:
"Analysis of Barrel-Aged Kentucky Bourbon Whiskey by Ultrahigh Resolution Mass Spectrometry" - Yang 2020
"The effect of cask charring on Scotch whisky maturation" - Clyde 1993
"Spectrophotometric determination of caramel content in spirits aged in oak casks" - Bosocolo 2002