Modular forms of higher weight are (certain) functions on $SL_2(\mathbb R)$ that transform according to a certain character of the non-split torus $SO_2(\mathbb R)$ of $SL_2(\mathbb R)$. So one expects their analogue to be (certain) functions on $SL_2(\mathbb F_q((t^{-1})))$ that transform according to a certain character of a non-split torus in $SL_2(\mathbb F_q((t^{-1})))$. There are multiple non-split tori to choose from. For example one can take the norm one elements of $\mathbb F_{q^2}((t^{-1}))^\times$. Let's say we choose a character of conductor expends $n$, i.e. that depends only on the element modulo $t^{-n}$.

In the classical case we have a holomorphicity condition on the functions, which has the consequence of forcing them to lie in an irreducible representation of $SL_2(\mathbb R)$. Similarly in this case we have to put a condition on the functions that forces them to lie in an irreducible representation. The right kinds of conditions to look for come from constructions of irreducible representations of groups over non-archimedean local fields. For simplicity take $n$ even. The construction I know is to consider the subgroup $J$ of $SL_2(\mathbb F_q[[t^{-1}]])$ consisting of all elements congruent modulo $t^{-n/2}$ to elements of the non-split torus of norm $1$ elements of $\mathbb F_{q^2}[[t^{-2}]]^{\times}$, and then take a one-dimensional character $\chi$ of $J$ that restricts to our fixed character of the torus, and consider functions $f$ on $SL_2(\mathbb F_q((t^{-1}))))$ that satisfies $f(gh) =f(g)\chi(h)$ for $h\in J$.

Finally we take our functions $f$ to also be left-invariant under a congruence subgroup $\Gamma_0(\mathfrak n)$.

The "weight" in this case is the conductor $n$ and the character $\chi$. We don't have a one-to-one bijection of weights because of the difference between local behaviors at archimedean and non-archimedean forms.

It's hard to look for an analogue of the modularity theorem because modularity statements about higher weight classical modular forms will usually be expressed using Hodge theory which doesn't exist over function fields. However, for analytic purposes, these kinds of forms will often behave like modular forms of higher weight. This is explained, in different language, in this paper on the sup-norm problem.