Can one tell the different sources of dashi by taste?

Sure, it's really not that difficult if you've actually tasted several kinds of dashi-jiru. It's more a matter of experience. There is decidedly a flavor to each category of dashi; it's not just "umami" or you would be able to get away with just throwing in a bunch of MSG into a bowl of water. But the flavor is mostly from aroma, like with other types of soup stock, since you haven't added any of the basic "tastes" other than the kombu and occasionally residual salt from the dried fish at the point the dashi is made.

At home, I often make a vegetarian one with kelp and porcini, sometimes with added cabbage; this the closest I've been able to come to the katsuo-dashi taste without actually using fish ingredients. It's a variation of a more standard Japanese kombu-dashi that's made with kombu and sometimes dried shiitake, but works for a broader range of dishes than the shiitake or minimalist kombu-dashi.

In practice, the main categories of dashi are these:

• Katsuo-dashi: Made with dried, cured skipjack tuna (aka bonito), and shaved. It has a slightly bitter taste. It is best made with the addition dried kombu. Tastes richer with either thicker shavings of katsuo-bushi or just-before-cooking shaving, when possible.
• Niboshi-dashi: Made with small dried whole fish (sometimes with heads removed, sometimes not, depending on the type of fish). Generally has a more pungent, richer flavor. There are several potential kinds of fish used and there was once substantial regional variation in this category, so the taste can vary quite a bit depending on the exact fish variety and how the fish was cured.
• Kombu-dashi: Made just with kombu, usually for adding a little bit of aroma and complexity to simple dishes, but rarely for soups, except for nabemono, where you'd typically have additional seafood or meats in the hot-pot. (I usually lump kombu-shiitake or kombu-porcini dashi in this category, but for no particular reason).
• Tori-dashi or gara-dashi: Made with chicken or other fowl bones. Except for fact that mirepoix isn't usually in the Japanese equivalent of this, so you won't have the celery base, it's similar to a typical Western-style chicken broth.
• Ton-kotsu-dashi, made with pork bones. Generally pretty hearty and full of fat.

Dashi is the analog to a soup stock, so it generally does not have added salt. Accordingly, you won't serve a straight dashi with an udon; you'd turn it into a broth, generally called "kakejiru". This will contain added salt, shoyu, and usually some combination of sugar, sake and/or mirin.

Before you put salt in it, you'll mostly "smell" a difference. After that, you can certainly at least distinguish between katsuo and kombu dashi, and with a little practice, you can distinguish between niboshi-dashi and katsuo-dashi. For me, the difference between katsuo-dashi and the instant "hon-dashi" is usually not subtle; the hon-dashi is a bit aggressive and harsh to my taste, but not everyone feels that way. (For what it's worth, I'm as close as practical to being a vegetarian for someone who regularly dines out in Japan). Tonkotsu-dashi and tori-dashi are somewhat rarely used in home cooking (though there are some exceptions in west Japan and Okinawa), but the taste difference is pretty obvious.

I actually tend to prefer niboshi-dashi over katsuo-dashi, but it depends on the type of fish. There's a dried sardine from Korea that I find a little harder to stomach when used in dashi.

I can't speak to allergies, but I suppose it's possible that different types of fish may cause some people to react differently. I suppose you'd have to ask an allergist.