Yes, there is definitely a performance difference between different types of kayaks. In fact, there are performance differences among the same types of kayaks. For example, a 14-foot kayak will always track more straightly than a 10-foot kayak. A wide recreational kayak will always have more primary stability than a narrower touring kayak.
Even if you both get the same type of kayak I suspect you'll find they work differently for each of you. No doubt you both do not weigh the same -- if you're heavier, a given kayak will sit lower in the water than for her. That may result in increased stability, or it could mean that you capsize easily. Alternatively, consider height and armspan: if she's short she may find it difficult to paddle in a wide rec kayak where the paddle is constantly bumping the edge of the boat.
In short: you should each look for a kayak that suits you. That means you will see performance differences between the two kayaks, but if the boats really do suit each of you, that won't be a problem because you've found something that fits well. As to a fishing kayak specifically, if you get a longer kayak than her you'll likely be able to keep up with her on a day trip (though at the expense of maneuverability).
Of course, keeping up with a partner is about more than the just the kayak. The paddle will also make a difference (especially its weight as it relates to your fatigue), and paddling technique can be a big difference between feeling good at the end of an outing and feeling whipped.