Meillassoux' Cogito shares the notion with the famous one from Descartes because it serves the same function in the sense that it is thought as providing a firm basis for all future philosophy/knowledge.
As @MauroAllegranza pointed out in the comments, Meillassoux additionally claims to have a "homologous procedure" in "proving" his cogito, i.e. a sceptical search for necessary and universal truth.
It is - according to Meillassoux himself - different in (at least) two ways: It is not representationalist (subject-object relation) and it is not solipsistic.
Although frowned upon, I think it is helpful here to blockquote the important part from After Finitude from the beginning of chapter three directly and comment it:
Following Descartes' example,
we are attempting to move beyond a 'cogito' by accessing an
absolute capable of founding science's (ancestral) discourse. But
the cogito in question is no longer the Cartesian cogito - it is a
'correlationist cogito' that encloses thought in a reciprocal relation
to being, one which is merely the mask for thought's underlying
relation to itself.
As Förster in his book The Twenty-Five Years of Philosophy points out in Chapter 6 and further develops in the following one, Kantian philosophy is kind of 'unfinished'. It lacks a monistic 'first principle' from which it could be derived. Without it, the critical project cannot be what it wanted to: Philosophy as a science. Hegel thought it as accomplished in 1806, but history showed that this openness to sceptical thought proved problematic.
One of the points of After Finitude is that because of this lack of a first principle waterproof against scepticism, philosophy (and humanity in general!) oscillated between critical, sceptical and dogmatic thought for the last 200 years.
Therefore, it is "cogito" in the sense that it should provide a firm ground on which all philosophy (or any knowledge) can stand thereafter as a science. This is exactly what Descartes wanted to provide in his Meditations and the cogito marked this exact achievement in his thought.
This cogito differs from the Cartesian cogito in at
least two ways:
Obviously, we can use these two points and wouldn't even have to resort to speculation...
- The correlationist cogito cannot necessarily be identified
with a metaphysics of representation, since it can be a function of
a conception of the correlation between thought and being other
than the one between subject and object (e.g. Heidegger's copropriation
of man and Being).
Descartes' Cogito is a fact of self-representation of the subject as its object. As a sidenote, it was Fichte's original insight - and not Heidegger's - that this form of representation is special (see Förster, p. 163 fn. 14). Meillassoux obviously thinks that his 'cogito' is not such a representationalist conception (something that works against Kant as well, btw).
- It is not strictly speaking a solipsistic cogito, but rather a
'cogitamus', since it founds science's objective truth upon an
intersubjective consensus among consciousnesses. Yet the
correlationist cogito also institutes a certain kind of solipsism,
which could be called a 'species solipsism', or a 'solipsism of the
community', since it ratifies the impossibility of thinking any
reality that would be anterior or posterior to the community of
thinking beings. This community only has dealings with itself,
and with the world with which it is contemporaneous.
One of the most common criticisms of Descartes' is his underlying solipsism (imho inevitable outcome of certain kinds of scepticism). Meillassoux seems convinced that all he could be objected for is his 'species solipsism', which obviously is no problem at all as long as we do not have an external standard of truth. His conception is more in line with en vogue versions of realism (although Descartes tried to account for that as well).
My personal opinion: I think that Meillassoux cannot answer the question how subjectivity (the subject-object divide) is possible at all (both genealogical and transcendental) - the main question for Kant's contemporaries after being left with the task to find a common foundation for his system. His whole project therefore kind of misses the (sceptical, fundamental) point of the problem it wanted to tackle in the first place.