  This work brings together the moment matching approach based on Loewner
functions and the classical Loewner framework based on the Loewner pencil in
the case of bilinear systems. New Loewner functions are defined based on the
bilinear Loewner framework, and a Loewner equivalent model is produced using
these functions. This model is composed of infinite series that needs to be
truncated in order to be implemented in practice. In this context, a new notion
of approximate Loewner equivalence is introduced. In the end, it is shown that
the moment matching procedure based on the proposed Loewner functions and the
classical interpolatory bilinear Loewner framework both result in
$\kappa$-Loewner equivalent models, the main difference being that the latter
preserves bilinearity at the expense of a higher order.
