There is no proper padding and the message $m$ can be recovered easily if $e$ is small.
First, we can rewrite $(m\mathbin\| m\mathbin\|\ldots\mathbin\|m)$ as
$$
(m + m2^{128} + \cdots + m2^{128\times 7}) = m(1 + 2^{128} + \cdots + 2^{128\times 7}),
$$
and when we put it back in the equation, we have
$$
c = m^e(1 + 2^{128} + \cdots + 2^{128\times 7})^e \mod n.
$$
We can compute
$$
c' = c\times (1 + 2^{128} + \cdots + 2^{128\times 7})^{-e} \mod n,
$$
so we have the relation $c' \equiv m^e \bmod n$. In the case that $m^e < n$ (which happens for $e=3$, $5$ or $7$), then this is in fact an equality:
$$
c' = m^e.
$$
The value $m$ can be recovered by taking the $e$-nth root of $c'$.