If you somehow ended up in these sewers, knowing the term "Azure Dragon's Inn" and its meaning will help a lot with survival matters, especially when connecting with other sewerfolks out there. Frankly, nobody knows where did this term came from or its origins. If you ask a sewerfolk, the most likely answer will be "I learned it from another person". If you hear someone claiming to know the origins of that term, the possibilities are that they are either lying or the person who told them the "origin story" lied to them. But the origin of it really doesn't matter as long as it doesn't loses its value in sewer communication and connection networks.
The term "Azure Dragon's Inn" indicates a certain meeting spot for sewerfolks, where they can safely settle down and rest for a while without facing any dangers or obstacles, while also meeting with other sewerfolks. The number of inns across the sewer is unknown, but they are usually found in much higher quantities in places that are safe to settle down. For example, you are more likely to stumble upon one in a place that has better air quality and a calmer environment. Those places are usually marked by the initials of the term, and you may also find some materials left by previous visitors if you're lucky (its important to note that leaving some of your inventory there for the future visitors before leaving is considered a courteous manner, but most people don't pay attention to that rule since there are some people present in the spot most of the time whenever somebody new arrives).