What's the name of the black-brown decoration sauce used in French cooking?
It is most probably crema di balsamico, a quite popular condiment, even often only used for decorative purposes. It can both be used with savory dishes, but also with sweet dishes, as in e.g. ice cream or gelato.

Traditionally, crema di balsamico is made by reducing grape juice and optionally wine to the point where the sugar in the grape juice starts to caramelise and then deglaze the reduction with balsamic vinegar. Convenience products are often pepped with food colouring and thickening agents.