On the level of elements, it doesn't really make sense to ask whether $\otimes$ is the algebraic or the topological tensor product. Here's why:
For Hilbert spaces, the topological tensor product is the completion of the algebraic tensor product with respect to the Hilbert norm. This means the algebraic tensor product is a dense subset of the topological version, so the tensor products of elements of the space is a tensor that lies in both the algebraic and the topological product.
If they claim that the tensor products of elements are a basis for the tensor product space, then one must ask in which sense they are a basis - i.e. are infinite sums of basis vectors allowed. Since the physicist usually allows this (implicitly, by talking about countable bases of infinite-dimensional spaces), we should assume the text means that. Allowing infinite sums, the tensor products of basis elements of the spaces that are tensored are such a (Schauder) basis of the topological tensor product (since that is defined as the completion, i.e. "adding limits" of finite combinations of the basis, which is what an infinite sum/series is).
Therefore, the Fock space that is constructed this way is intended to be the topological tensor product.