|Victoria sponge: technically a sandwich? A defense.|
First, we need to agree on a definition of 'bread'. The OED says: "A well-known article of food prepared by moistening, kneading, and baking meal or flour, generally with the addition of yeast or leaven." Wikipedia claims: "Bread is a staple food prepared from a dough of flour and water, usually by baking." So I think we can reasonably say:
X is bread if and only if X is a baked mixture of flour and water.
So, pace the above responses, tortillas are in fact breads, as are matzo, naan, crackers, Eucharist wafers, injera, etc. This also implies that cakes are breads, q.v. below (and cf. again the OED: "As name of an object, with plural: A baked mass of bread or substance of similar kind, distinguished from a loaf or other ordinary bread, either by its form or by its composition"; hence, cakes are a kind of bread.)
Now, it does seem to me plausible to say that Y is a sandwich only if Y includes bread: it's a necessary condition for being a sandwich. However, I don't think bread is a sufficient condition—it is not true that if Y includes bread, then it is a sandwich, because then bread puddings, Thanksgiving stuffing, and salads with croutons would be sandwiches.
This raises the question of how we define "sandwich." Here, Wikipedia insists that there must be two slices of bread: "A sandwich is a food item consisting of one or more types of food placed on or between slices of bread, or more generally any dish wherein two or more pieces of bread serve as a container or wrapper for some other food." This would seem to exclude the tortilla wrap. But we now have problems of insufficient definitional capture: our definition seems to exclude a number of things that are sandwiches, like the tartine and the hot turkey sandwich. Cp. the addendum to the OED's definition:
An article of food for a light meal or snack, composed of two thin slices of bread, usu. buttered, with a savoury (orig. spec. meat, esp. beef or ham) or other filling. Freq. with specifying word prefixed indicating contents, as ham sandwich, egg sandwich, watercress sandwich, peanut butter sandwich (see peanut butter n.), or form, as club sandwich (see club n. Compounds 3), Dagwood sandwich, Denver sandwich, hero sandwich (see hero n. Compounds 4), poor boy sandwich (see poor boy n.), submarine sandwich (see submarine n.). Occas. with only one slice of bread, as in open sandwich or open-faced sandwich (see open-face adj. 2), or with biscuits, sliced buns, or cake.
This is more correct, because it captures the tartine, the hot turkey sandwich, and indeed, the wrap. But is our definition now too inclusive? After all, we now include as "sandwiches": tacos, burritos, Pop-Tarts, Belgian waffles, et al. Rather than succumb to the temptation to make "sandwich" a notion based on Familienähnlichkeit, I think that we simply need to accept that tacos and the like are indeed kinds of sandwiches. Cf. definition of "sandwich" above (viz., "with a savoury... or other filling") with the OED's sensible definition of taco: "A Mexican snack comprising a fried, unleavened cornmeal pancake or tortilla filled with seasoned mincemeat, chicken, cheese, beans, etc." Since we have agreed that cornmeal pancakes and tortillas meet our definition of "bread," and you only need one piece of bread for Y to be a sandwich, it follows that tacos are sandwiches. Further, both falafel in a pita and quesadillas are sandwiches as well.
This does imply in addition that many well-known cakes are, in fact, sandwiches. A Victoria sponge is technically jam and cream sandwiched between two cakes. Indeed, it is possible that any cake bearing frosting is in fact a sandwich. Some will object that this is an obvious instance of a definition being too broad. To which I reply simply that no definition that allows you to eat Oreos and birthday cake for lunch and claim that you're having a sandwich is an altogether bad definition. (Oreos, I think we can all agree, are certainly sandwiches.)
[*An addendum: pies are technically sandwiches as well. An odd conclusion for an American, to be sure; but less so—indeed, sensibly—if you've spent time in Britain or the Upper Peninsula of Michigan, where handheld savory pies ("pasties," pronounced with a short A, as opposed to the other kind) are an important part of the lunch-option landscape.]