Is synapsida both a class and a clade?

That depends on your definition of a clade. Synapsida is usually used to refer to a group of primitive reptiles that evolved very early even earlier than the diapsids (snakes, lizards, crocodilians, and dinosaurs). One of the synapsids was the ancestor of the therapsids, and a therapsid eventually evolved into the ancestor of the mammals.

Few people classify Synapsida as a class under the Linnean system. Synapsida is usually classified as a subclass within class Reptilia, and Therapsida is usually classified as an order within Synapsida. Since every taxon must be monophyletic, meaning they must share a nearest common ancestor with one another, all taxa must therefore be clades, notwithstanding the inevitable few cases in which (despite our best efforts) organisms that are only convergently similar are lumped into the same polyphyletic group by mistake. However, the cladists hijacked the terms clade and monophyletic by narrowing their definitions to require that all descendants of the common ancestor, and the common ancestor itself must also be included. Since Synapsida does not include any mammals, the cladists would claim that it is not a clade, but most Darwinians would disagree and say that it is.

A cladist proposes to make all synapsids part of a clade sensu cladism, so they combined the synapsids, therapsids and mammals into a single taxon known as "Mammalia." That means they are calling synapsids and therapsids "mammals." That is of course absurd, because mammalian features such as hair and mammary glands almost certainly were not present in the group of synapsids known as pelycosaurs. It is also very doubtful that most therapsids (especially the dicynodonts) had hair or mammary glands. Not surprisingly, most people do not accept this cladistic re-definition of Mammalia and they continue to classify synapsids and therapsids as reptiles and mammals in a different class Mammalia.

In evolutionary terms, the mammals are entirely within the Synapsida. Their status as a separate class is traditional Linnaean taxonomy: as a clade they are a sub-group of the Therapsida.

Can you be more specific please?