Watco does not list its formula, but it is likely similar to homebrew mixes of varnish, oil, and solvent. Once cured, the varnish/oil has polymerized into a soft varnish finish.
I have cleaned the cured accumulation with commercial brush cleaners; the kind sold for removal of hardened gummy finishes in brush bristles. Also useful is the drain cleaner sold at the Dollar store (lye). If you use lye, eye and skin protection from lye solutions is absolutely needed. Lye solution will dissolve natural bristles, but would not be an issue with a plastic bottle.