Cross-reactivity refers to the action of antibodies. Cross-reactivity arises because antibodies aren't always 100% specific for their antigens.
Say gluten has the molecular appearance of an elephant - a trunk, 4 stocky legs, a tail, etc... when our bodies produce antibodies that recognize gluten, they may only produce an antibody that recognizes one part of gluten. Perhaps just tail, leg, or trunk. If you made the antibody against the trunk, you'll have a very specific antibody, as no other animals have trunks. If you made an antibody against the stocky leg, you might also react with something like a rhino. Or an antibody against the tail, might also recognize all sorts of other animal tails.
All gluten antibodies are not necessarily the same. Some may be more or less specific than others. It all depends on what molecular feature the antibodies recognize. Some are specific to gluten, others may be similar between various proteins. This explains why some gluten-intolerant folks cannot eat corn, but some can. Some can eat rice, but others cannot. Some react against casein, but others tolerate casein just fine.
Cross-reactivity also explains why leaky gut causes auto-immunity. You might create antibodies against wheat that also recognize your thyroid or pancreas, your body then initiates an immune response against your thyroid or pancreas.
So yes, what the doctor is saying is true: cross-reactivity might continue causing problems even when you eliminate what you're sensitive to. Of course, how do one know if you're simply cross reacting to corn due to a wheat sensitivity or you're simply also sensitive to corn primarily. That probably requires some antibody testing and blood work, though I don't know anything about that.