It is a descriptive tag (as are all tags to some extent), it does not affect supportability.
Exsanguinate, mechanically, can chain.
Because of this, the Chain Support can support it - its supportability text says "Supports projectile skills, and any other skills that chain.", and Exsanguinate is in that second group. The tag has no effect on this.
Because Exsanguinate is a (non-projectile) skill that can chain, it is given the Chaining tag, primarily to make it clear to users that the skill is capable of chaining (I don't think we give out any bonuses to gems based on having the Chaining tag and I hope we never do because then we'd probably have to go back and add it to all the projectile gems that are currently assumed to be able to chain by default).
The supportability and the tag do not affect each other. Both are there because the skill works in a way that allows it to chain. The tag is not needed for supportability, the supportability is not nee...
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