Is there a name for the taste coming from alkaline food?

There is no single word to describe that taste, and no single taste receptor for alkaline substances either (whereas there are taste receptors for acids).

The most recognizable taste to encounter would be "soapy". You can see it as a bit of a special case though, since it is not the taste of a pure alkaline substance, but the taste you get when the alkaline substance reacts with fats. Since most food contains fats, this is what you frequently get if you try to make alkaline food.

You are also likely to get descriptions like "chemically tasting", since it is not a flavor we frequently encounter in food, but it can remind us of the aroma of some inorganic substances in our daily lives. Try overdoing double-acting baking powder in a muffin, or making it with ammonium or potassium based leaveners instead of the widespread sodium ones, and you'll see what I mean.

Beyond that, we don't really have good descriptors. And I am not even sure pretzels count - the goal there is to have the lye react away with the gluten, so they might not be very alkaline in the end. Also, the salt on the surface and the unusual texture distracts us a lot from the subtler flavors there.

The question intrigued me, mostly because I was under the impression that foods just varied in the degree of acidity, and we don't eat alkali materials. So looking up on a couple of charts found randomly on the internet, ...

Approximate pH of Foods and Food products
pH Values of Common Foods and Ingredients

... and keeping in mind Mr. Janowski's statement from 10th grade that water has pH 7, and anything higher than that is a "base", I find that yes, we do eat alkali foods [he also taught that "alkali" was a fancy word for "base"], but not very many. Some of the ones that jumped out at me were:

corn, ripe olives, tofu, birds nest soup[??], clams, coconut, conch (pH of 7.52-8.40!!), graham crackers, grass jelly[??], hominy, lobster, soda crackers, cooked spinach. (Some of those were listed with a range of pH values extending from just below 7 to just above 7). So what taste do they have in common? In fact what do they have in common at all? Many have either been heavily processed maybe with alkaline chemicals (ripe olives, tofu, hominy), or come from the sea (conch, lobster, clams). Those examples seem to mostly have a salty component to their flavor. As for the land-origin, naturally occuring examples, i.e. corn and coconut, they aren't salty, and they don't seem to have any taste in common to each other or those other basic foods.

So my (disappointing) answer is, I don't think you can taste the "basic" quality itself, except that many foods which are basic are often salty because of processing or being borne of sea water (pH 8.1 - 8.2 per internet).

Alkaline in food does not impart any specific named "taste" to the food. So the food is just being described as alkaline or basic. Unless you are using food texture to describe alkaline foods, as foods treated with alkaline usually ends up with a different texture such as being firmer/denser.

On a related note:

Even though there is not much flavor from adding lots of alkaline to food, what it does do very well is it drastically changes the texture of food. Alkaline(Lye) treatment usually gives the food items a more slimy gelatinous texture before you cook it.

Lye water is used commonly in Asian cuisine, it makes food alkaline. Powdered Lye is also commonly used to make pretzels.