Why do hotels (apparently) require you to leave and then ask for your room key back at the reception when leaving/returning to the hotel? [duplicate]

This is seldom the case nowadays but it still does happen. You are absolutely right that the practice is silly and is generally unnecessary. Your question about international hotels covers an extremely wide net that includes vastly different practices across locations and property types. So, keep in mind that there are vastly different protocols and not one procedure for all international hotels but generally:

• When arriving at a hotel, you nearly always check-in. At this point you show either a reservation or ID and staff hands you one or more keys to your room. This confirms who you are, the duration of your stay and your payment agreement (here too there are many ways this works - topics for other questions).
• You or anyone in your party can generally go in and out of your room as you please but you normally have to leave the room locked to discourage opportunistic thieves. While you are out, cleaning staff comes in to clean the room, usually but not always daily. Most hotels have a master key or a duplicate of each room key, so they can enter while you are away. However, there are some places, usually small hotels in poor countries where copies are expensive and so they ask you to leave the key for use by staff. This also has the secondary effect of letting them know when you are away.
• When you go back to your room during your stay, you need your key to go back in. If you left it at the desk, you have to ask for it and indeed they hand it extremely easily! You generally only need to state your name of even just the room number and they give it to you. This always seems insecure to me and I often just tell them that I'm keeping the key with me. Staff rarely objects unless it is because they need to key to clean the room and in those cases, I only leave it with them during one outing.

Now, in small hotels, it seems that they do this by memory but that may be the case only some of the time. Evening and daytime staff are not always the same, as for weekday and weekend staff and so there are times when you will leave your key with someone and get it back from someone else. The insanity is that not once have they ever asked me for ID and I've stayed in several hundred hotels in 63 countries.

This protocol would be highly unsustainable in large hotels and I have only seen it in places with fewer than 12 or so rooms. Many modern hotels now generate a magnetic key when a guest checks in and it gets invalidated when you check-out. People usually hand them to the desk to get erased. These keys can contain information identifying the customer and so should be wiped or destroyed to avoid privacy issues.