  A new set of multiple orthogonal polynomials of both type I and type II with
respect to two weight functions involving Gauss' hypergeometric function on the
interval $(0,1)$ is studied. This type of polynomials have direct applications
in the investigation of singular values of products of Ginibre matrices, in the
analysis of rational solutions to Painlev\'e equations and are connected with
branched continued fractions and total positivity problems in combinatorics.
The pair of orthogonality measures is shown to be a Nikishin system and to
satisfy a matrix Pearson-type differential equation. The focus is on the
polynomials whose indexes lie on the step line, for which it is shown that
differentiation on the variable gives a shift on the parameters, therefore
satisfying Hahn's property. We obtain a Rodrigues-type formula for type I,
while a more detailed characterisation is given for the type II polynomials
(aka $2$-orthogonal polynomials) which include: an explicit expression as a
terminating hypergeometric series, a third-order differential equation, and a
third-order recurrence relation. The asymptotic behaviour of their recurrence
coefficients mimics those of Jacobi-Pi\~neiro polynomials, based on which,
their zero asymptotic distribution and a Mehler-Heine asymptotic formula near
the origin are given. Particular choices on the parameters degenerate in some
known systems such as special cases of the Jacobi-Pi\~neiro polynomials,
Jacobi-type $2$-orthogonal polynomials, and components of the cubic
decomposition of threefold symmetric Hahn-classical polynomials. Equally
considered are confluence relations to other known polynomial sets, such as
multiple orthogonal polynomials with respect to Tricomi functions.
