Typical phone jacks and Ethernet jacks look the same from a distance. They are slightly different sizes and the key is the number of pins.
- 2 - single phone line and some digital (but not VOIP) phone systems
- 4 - two phone lines and some phone systems (analog, digital or hybrid)
- 6 - some phone systems, some terminals from ye olden days
- 8 - Ethernet networks, some terminals from ye olden days
As far as I can tell the upper jack of the 2-jack faceplate has 6 wires. Which will not work for Ethernet. I can't tell what the others have. But all is not lost - you may have standard 4-pair cables even if the jacks have fewer than 8 pins. If that's the case then the quality (old junk vs. Cat 3 vs. Cat 5 vs. Cat 5e) will determine whether you can make use of the cables or not.
But in addition to the number of wires, the big issue is the routing. For traditional phone service (analog voice, fax and modem) you can use a star configuration or daisy-chain. Twisted pair Ethernet networks and most phone systems require a star configuration. If you either have a cable (8-wire, preferably Cat 5 or better) going directly between the two locations you want to connect or if all the cables go to one central location then you can use the cables to set up an Ethernet network.
Based on the pictures, it looks like the wires attached to each jack are a pair from a 3-pair cable. While it is possible to run Ethernet on 2 pairs, generally speaking once twisted pair Ethernet became common the usual thing to do was to wire everything with 4-pair cable and mix/match as needed for various phone systems, Ethernet, etc. So a 3-pair cable would generally predate the Ethernet era or was used to save money, in which case it would almost certainly not be even Cat 3 quality.
You might be able to use the old cable to pull new cable. That will depend on how it was installed, which is very hard to tell from the boxes. Testing first to see if any of the wires are connected, and searching to see if they are going to a central location, are the next steps.