In US English, the phrase "coin bank" or "change bank" fits your description. You can verify this by doing a Google Images search on "coin bank." Very commonly, people in the US who are simply saving loose change save it in a "change jar," which is just any available jar the right size. Money that people keep in coin banks and change jars, which is thus not in circulation, actually represents a source of profit and revenue to the government. Your coin bank is a interest-free loan you are making to the government. If you find this curious or interesting, do a search on the obscure, technical word "seigniorage," or read the Wikipedia article https://en.wikipedia.org/wiki/Seigniorage (There is also a "cash box," which is different. A "cash box" has no slot. It is often a grey metal box. It has a hinged lid, a latch and a lock of some kind. The inside usually has drawers for organizing paper money and coins. It is used when people are dealing with small amounts of cash and are not a regular business with a cash register. A garden club might keep member dues in one, an office might keep "petty cash" in one.)

When I was a kid, my brother and I had a small box with a slit in it where we kept coins. We simply called it a bank. I agree that "coin bank" would be an acceptable term for it.