Help in identifying a thestery wall socket

This really shouldn't be an answer since Chris H.'s answer nailed it (and should be accepted), but I want to address several comments from several answers, and this is far too long for a comment on any of them.

1. Since your modem doesn't seem to have come with screws and drywall anchors, you'll have to find something
   1. You are now looking for hardware, not electronics, so do not go to the electronics store to find a screw, go to a hardware store. Also, don't bother with trying to order them online. You only need 2 and the few cents, fractions of a Euro (sorry, don't know the correct term), whatever, that you save will not make up for the time you wait for them to be shipped or the annoyance of discovering that they don't fit well.
   2. It seems you're in Finland. I've never been there (would love to visit some day!) so I don't know what you have in the way of hardware stores, but here in the US, we have massive hardware stores with aisles full of screws of varying types. I would suggest you take the modem (without its box) to the store (stop at the front desk if necessary to show them you're bringing it in with you, though they probably don't sell modems at the hardware store & won't question you when you walk out without paying for it) then wander down the screw aisle trying different screws until I found one who's head fits nicely into the hole in the back of the modem and who's shank will slide into the slot, then buy a small package (or just 2 if you can get them individually).
   3. Side note: having a few extra screws from this project is a nice start to the collection of random screws that you'll need to make it through life - these types of little projects crop up all the time and a jar of screws to sort through to find something that's close enough to get the next random small job done is very satisfying!
   4. The screw heads don't have to be a precise fit. Actually if they're too snug, they'll make installing the modem on the wall very difficult. You want a bit of play around the screw head so you can easily slide the modem on once the screws are on the wall.
   5. The screws you show in your picture look like they'll work just fine, especially if you install them into a stud behind your drywall. (See 2.2, below).
   6. Pan-head screws would be ideal, with their relatively thin head and flat bottom, however a wood screw (with a sloped bottom) will work just fine. I'm more likely to end up with a wood screw (from my random collection of extra screws from previous projects) for this type of hanging assignment than a pan-head, and I've never had anything fall off a wall. Actually, as light as the modem is, pretty much any screw will work if the head & shank fit the slot you've got.
   7. The one screw you do not want is a machine screw with a flat bottom. These are designed to go into pre-tapped threads and will not cut their own threads very well into drywall, studs, or the side of your desk.
   8. If you're installing onto the side of your desk, you'll want much, much shorter screws with threads all the way to the bottom of the head, since your desk panel is much thinner. (Those pictured will stick all the way through the side of the desk and snag your leg. Every. Single. Time. Guaranteed!)
2. If you're installing this onto drywall, you'll want to either:
   1. Drywall anchors which will allow you to put the modem anywhere you want. These usually come with screws sized to fit the anchors. Or,
   2. Get some longer screws (1.5-2", approx 35-50mm) and make sure they go into a stud behind the drywall.
3. Once you've got your screws:
   1. Use one of the many, all valid methods mentioned in the comments on Chris H's answer to locate the screws on the wall.
   2. Pre-drill holes if you're going into the side of your desk or into a stud or using a wall anchor.
   3. If you're not using a wall anchor in drywall, you'll probably be fine. The modem itself weighs next to nothing and won't put a huge strain on the drywall. The biggest issue will be the additional weight of the cables (power & network) hanging out of it (mitigate this by hanging it lower to the ground, though, if it's also a router, that could impact your WiFi range), and you wiggling it around (it'll happen) as you insert cables.
   4. Insert your screws into the wall leaving 1/4 - 3/8" (6-10mm) of thread screw exposed.
4. Now the fun starts!
   1. You'll have to fiddle around a little bit to align the heads of both screws with the hanger holes at the same time, then slide the modem down (gravity is your friend!) so the screws move into the slot part of the hanger.
   2. If the modem is pretty snug to wall and doesn't wobble too much (how much? Don't know, it's your modem you get to decide when it's secure enough), you're done. Plug in your cables, grab a drink and dive into your favorite game/movie/whatever.
   3. If the screw heads won't get into their slots at the back of the modem at all, grab your screwdriver, back the screws out a turn (or so), go back to step 4.1, then repeat until you meet the criteria in step 4.2.
   4. If the screw heads go into the slots, but the modem wobbles excessively, the feet are nowhere near the wall, and all-in-all, you have the distinct feeling that it will fall off if someone in London sips his tea without his pinky raised, then unhook it, grab the screwdriver, turn the screws in a bit, then return to step 4.1.
   5. Repeat the steps above until you're successfully at 4.2. Note that you do not have to have a precise, exact fit of the feet against the wall for this to be securely attached. As a matter of fact, you'll want a couple of mm of play to make it easier it install (and remove at some point in the future, modems do die). I've used hanger systems like this for anything from a power strip to the leg of a desk to a large nick-knack shelf on the wall with tons of my wife's display items. None of them are super snug, nor are they excessively wobbly and none has ever made good an unplanned escape. Close enough is your guiding principle here.

If things go well for you, you'll do step 4.1 and everything will be perfect. Odds are, you'll be like me and you'll alternate between 4.2 & 4.3 a couple of times until you get it just right. That's just the way this game works.

One sure option is to cut the wires off immediately adjacent to the socket. Then you strip off a short bit of additional insulation on each wire to expose the electrical conductor inside.

It looks like that socket used to also provide a hook from which the cordage or swag chain of the old lamp would hang. You will have to carefully evaluate how the new lamp is secured. If it is a hanging unit you cannot simply let it hang from the electrical connections. If the lamp is a ceiling mounted fixture then make sure to hang it in a Code compliant way so that it is safe and does not put a strain on the electrical connections.