I never said we keep that kind of stuff in our banks all that often.
However that does not alter the fact that banks do offer safety deposit boxes where you CAN keep anything stuff you like - it is just a service people most commonly use for valuables like jewellery, antiques, sentimental items etc.
IT's also not a fact that no-one keeps that stuff in safety deposit boxes - there'll be plenty of rare, valuable or sentimentally valued books lurking in safety deposit boxes and I'm sure among the billions of people in this world there are at least one or two overly rich people who'd stick a DVD, Blu-ray or video game in a safety deposit box for some odd reason. Heck with video games it may not even be that odd a reason since some of the 20th Century gaming consoles and their games are becoming increasingly rare and increasingly valuable.
Given the historical nature of the Runescape world vs real world culture it makes perfect sense - they don't have electronic money, or debit cards they deal entirely in things that physically exist. Slap on the fact the society of Gielinor would be considered more of a bartering goods for goods society than a cash for everything society and banks would be opening to storing a variety of valuable trade goods as forms of currency outside of just gold. Then wedge in the safety deposit box notion and they'd gladly being storing our valuables for us too.
The thing is with a game like Runescape just because it doesn't conform perfectly to what we in the modern world do and how our institutions work does not mean it doesn't make sense.
The use of the word Bank refers to services and formats more commonly used in the past before we moved away from bartering and into electronic currency etc. but there are still instances of that service remaining in safety deposit boxes.
I mean if your gonna claim banks make no sense because of modern standards without any consideration of what came before and the other services they provide then you could claim almost nothing in-game makes sense as it doesn't fit modern standards,
People walking around armed to the teeth with swords etc - People don;t do that now, doesn't make sense.
People being freely allowed and encouraged to slaughter rare and sentient animals - doesn't male sense, animal rights groups would stop it.
People relying on wood fires and stoves for cooking - nope no sense there electric and gas cookers everywhere these days.
The list could go on. The fact is you'll never find much sense comparing Runescape purely to modern society alone, you have to consider historic societies and what things meant/were in their cultures.