Wrapping fig tree
A thick fleece blanket may absorb a lot of water and become very heavy, but it depends upon how you plan to use the blanket. Wrapping it around the base of the tree, it may be like a big sponge and keep the trunk too soaking wet all the time. If a layer of dead leaves or straw is kept in place with the blanket, that should be fine.
Read over the paragraphs about overwintering on our fig webpage: https://extension.umd.edu/hgic/figs The main objective is to prevent the branches from freezing, so lots of insulation is good.