The secret to your grandmother's canja is probably all in the cooking time, because unlike what a lot of people do quickly, traditional canja needs hours to develop that rich, full-bodied broth. Take a whole chicken (or the meatier parts if you prefer), put it in a large pot with cold water, bring it to a boil and remove that foam that comes up at the beginning. Then add a whole onion (yes, with the skin on is better, it gives color), about 2-3 peeled garlic cloves, carrot sliced into thick rounds, a celery stalk if you have it, salt and let it simmer on low heat for about 2-3 hours at least. The longer it cooks, the better the broth becomes.
About an hour before you're done, take out the chicken, shred the meat and put just the meat back in the pot, then remove the big vegetables (they've already given everything they had to the broth). This is where the rice comes in - usually you use regular round rice (like carolino), add about a cup or so and let it cook until it's really soft and releases a starch that makes the soup naturally creamy. Some people add a whole beaten egg at the end, stirring well to create those egg threads, which gives it a super characteristic texture. The difference from a more "quick" soup is exactly that: the patience of letting everything cook slowly creates a deep broth that quick canja can never achieve.