Fibers of fibrations of a 3-manifold over $S^1$

There are simple examples with $M = F \times S^1$ for $F$ a closed surface of genus $2$ or more. Choose a nonseparating simple closed curve $C$ in $F$, then take $n$ fibers $F_1,\cdots,F_n$ of $F\times S^1$, cut these fibers along the torus $T=C\times S^1$, and reglue the resulting cut surfaces so that $F_i$ connects to $F_{i+1}$ when it crosses $T$, with subscripts taken mod $n$. The resulting connected surface is an $n$-sheeted cover of $F$ and is a fiber of a new fibering of $M$ over $S^1$. The monodromy of this fibering is a periodic homeomorphism of the new fiber, of period $n$.

As you suspect, the answer is no. There are $3$-manifolds that fiber over the circle in infinitely many ways, with fibers of unbounded genera.

Thurston constructed a seminorm on $H_2(M,\partial M; \mathbb{R})$, now called the Thurston norm. The unit ball is a polyhedron, and Thurston showed that fibers of fibrations of $M$ over the circle are those integral classes which lie in the open cone on a collection of certain top dimensional faces (the ``fibered faces") of this ball. The norm is defined by extending the absolute value of the Euler characteristic of integral classes, and so any fibered manifold whose second homology has rank at least two gives you an example.

EDIT:

For a discussion of some explicit examples, see:

Hilden, Lozano, Montesinos-Amilibia, On hyperbolic 3-manifolds with an infinite number of fibrations over $S^1$. Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 1, 79–93.