Prepare chiles. Cut dried chiles open to remove seeds, then briefly toast in a dry skillet. Cover chiles with hot water and soak until rehydrated, about 10–30 minutes.
Make the marinade. Drain chiles. In a food processor or blender, combine rehydrated chiles with diced onion, garlic, vinegar, tomatoes, and ground spices and puree until smooth, adding a little water if needed. Taste and adjust seasoning.
Braise. Preheat the oven to 350°F. Transfer Dutch oven (with lid or aluminum-foil cover) to oven and braise until meat is falling-off-the-bone tender, about 4 hours.
Shred the meat. Remove the meat from the braising liquid. (Do not discard.) When cool enough to handle, shred the meat off the bones and set aside.
Assemble. For soup (birria con consomé), add 3 cups water to the braising liquid and stir to combine. Add the meat back into the soup and bring to a simmer. Ladle soup into bowls and serve the soup with warm corn tortillas, cilantro, and lime. For birria tacos, reduce the braising liquid to a thick sauce. Fry the shredded meat until crispy and briefly dip warm corn tortillas in the sauce. (You can crisp the meat and warm up the tortillas on the same hot griddle.) Fill sauced tortillas with crispy goat meat and top with cilantro and lime.