I found this reference, where the authors deal with the products you asked for.
EDIT The reference is
A MAPLE program for calculations with Schur functions
by M.J. Carvalho, S. D’Agostino
Computer Physics Communications 141 (2001) 282–295
From the paper (p.5 chap. 3.1 Multiplication and division of $m$-functions):
Let’s define the result of the addition
and subtraction of two partitions $(\mu_1,\mu_2, . . .)$
and $(\nu_1, \nu_2, . . .)$ as being the partition whose parts are
$(\mu_1 ± \nu_1,\mu_2 ± \nu_2, . . .)$. For these operations to be
meaningful, it is necessary that both partitions have
an equal number of parts; if they do not, then one
increases the number of parts of the shortest one by
adding enough zeros at the end. ... The multiplication (and division) of two m-functions
are then defined as
$$
m_{\alpha} m_{\beta} = \Sigma I_{\gamma}m_{\gamma}
$$
and
$$
m_{\alpha}/ m_{\beta} = \Sigma I_{\gamma'}m_{\gamma'}
$$
where the partitions $\gamma$,$\gamma'$ result from adding to or subtracting, respectively, from $\alpha$ all distinct partitions
obtained by permuting in all possible ways the parts
of $\beta$. Clearly, all $m$-functions involved
are functions of the same $r$ indeterminates, i.e. have
the same number of total parts.
The coefficient $I_\nu$, with $\nu = \gamma$ is given by
$$
I_\nu=n_\nu \frac{\dim (m_\alpha)}{\dim (m_\nu)}
$$
where $n_\nu$ is the number of times the same partition
$\nu$ appears in the process of adding or subtracting
partitions referred to above.
As far as I read, they don't give a special name to these coefficients.