I've been to several restaurants, usually sushi oriented, where I will sit down and order a drink (alcoholic), then within a few minutes of receiving my drink, the chef will either walk out and personally serve me a dish I didn't order, or if I'm sitting at the (sushi) bar, he will plop a dish in front of me and explain to me what it is (even though I have demonstrated that I don't speak a lick of japanese).
The dish is what I would call an appetizer; it's like minced sushi fish with sauce, or a small sticky fungus dish, or like a soup with daikon and tofu and what the chef called "pudding".
Usually I smile and nod/half-bow and repeatedly say "arigato thank you" at any brief pause while he's talking.
Also, most times I don't actually order any food, I just get drinks. This situation exclusively happens for my after work drinks where I'm heading elsewhere to get actual dinner meal. Is it rude to get served the dish I didn't order, consume said dish, and then leave (having only ordered drinks)? I don't think I've ever been charged beyond the price of the drinks and tax and service charges.
So my question is, is this a common Japanese custom or practice? If so, what is the name of this custom? What is the meaning behind it? And what is the proper response?