This phenomenon is called "cosmetic surgery."

If $K$ is an amphichiral knot in the $3$--sphere with exterior $M_K$, then $M_K(p/q) \cong - M_K(-p/q)$. So if $p/q$ is a hyperbolic filling slope, then this is the example you seek. Note that the homeomorphism is orientation-reversing. If there is an orientation-preserving homeomorphism between $M_k(p/q)$ and $M_K(p'/q')$, the surgeries are called ``truly cosmetic,'' and it is conjectured that cosmetic surgeries on hyperbolic knots are never truly cosmetic.

Take a look at Bleiler, Hodgson, and Weeks, ``Cosmetic surgery on knots,'' in
Proceedings of the Kirbyfest, pp. 23-34 , Geometry
and Topology Monographs, Vol. 2, Coventry, 1999 as a beginning reference, and the conjecture that cosmetic surgeries on hyperbolic knots are never truly cosmetic.

EDIT: Regarding your second question, I'm pretty sure there are hyperbolic manifolds $M$ that admit no cosmetic surgeries at all. Here's a sketch:

I'll use $\partial M$ to mean a cusp cross section of a $1$-cusped hyperbolic manifold $M$, normalized to be a flat torus of area one.

By a theorem of Nimershiem, the shapes of flat tori appearing as cusp cross sections of 1-cusped $3$-manifolds is dense in the moduli space of tori. (see Nimershiem, ``Isometry classes of flat 2-tori appearing as cusps of hyperbolic 3-manifolds are dense in the moduli space of the torus,'' in Proceedings of Low-Dimensional Topology, 1992, 133-142.)

By Nimershiem's theorem, we may pick an $M$ such that $\partial M$ has a very short curve $\gamma$ and so that the group of isometries of $\partial M$ up to isotopy is trivial. Let us also assume that $M(\gamma)$ has no short geodesics (you should be able to achieve this by some covering space tricks).

Now, suppose that $M$ admits cosmetic fillings $M(\alpha) \cong M(\beta)$ with $\alpha \neq \beta$. If $\alpha \neq \gamma$, then $\alpha$ is really long in $\partial M$, and it follows from sharp versions of the Hyperbolic Dehn Filling Theorem due to Hodgson and Kerckhoff that the core of the filling torus is short in $M(\alpha)$. This means that $\alpha \neq \gamma \neq \beta$. Furthermore, it means that the cores of the filling tori in $M(\alpha)$ and $M(\beta)$ are the unique short curves in these manifolds, respectively. So any homeomorphism $M(\alpha) \to M(\beta)$ must preserve these cores (this part of the argument is in lemma 1 of the Bleiler-Hodgson-Weeks paper). So, the homeomorphism $M(\alpha) \to M(\beta)$ comes from a homeomorphism (and hence an isometry) of $M$, which induces an isometry of $\partial M$ which is not isotopic to the identity (since we assume that $\alpha \neq \beta$). Since we have chosen $\partial M$ to have no such isometries, we have a contradiction, and must conclude that $M$ has no cosmetic fillings.