  We study the algebra of certain $q$-series, called bi-brackets, whose
coefficients are given by weighted sums over partitions. These series
incorporate the theory of modular forms for the full modular group as well as
the theory of multiple zeta values (MZV) due to their appearance in the Fourier
expansion of regularised multiple Eisenstein series. Using the conjugation of
partitions we obtain linear relations between bi-brackets, called the partition
relations, which yield naturally two different ways of expressing the product
of two bi-brackets similar to the stuffle and shuffle product of multiple zeta
values. Bi-brackets are generalizations of the generating functions of multiple
divisor sums, called brackets, $[s_1,\dots,s_l]$ studied by the author and U.
K\"uhn. We use the algebraic structure of bi-brackets to define further
$q$-series $[s_1,\dots,s_l]^{sh}$ and $[s_1,\dots,s_l]^\ast$ which satisfy the
shuffle and stuffle product formulas of MZV by using results about
quasi-shuffle algebras introduced by Hoffman. In a recent work the author and
K. Tasaka defined regularised multiple Eisenstein series $G^{sh}$, by using an
explicit connection to the coproduct on formal iterated integrals. These
satisfy the shuffle product formula. Applying the same concept for the
coproduct on quasi-shuffle algebras enables us to define multiple Eisenstein
series $G^\ast$ satisfying the stuffle product. We show that both $G^{sh}$ and
$G^\ast$ are given by linear combinations of products of MZV and bi-brackets.
Comparing these two regularized multiple Eisenstein series enables us to obtain
finite double shuffle relations for multiple Eisenstein series in low weights
which extend the relations proven before.
