Its definitely possible however it will be more of an "advanced" setup. Typically people can go wire for wire and have these install in less then 30 minutes. I filled holes and repainted the wall behind the thermostat and it took me 45 minutes.
You are missing a picture of the boilers thermostat though.
The second picture is showing Rc (24volt cooling transformer), a white wire connected to Y (cool relay) and G (fan relay). The W (heat relay) is the only terminal on that thermostat that controls the heat which is has nothing connected to it so there is another thermostat for the Buderus in the house.
The first picture is the boilers primary control. Although that isn't the most modern boiler rarely does a boiler use the TT (thermostat terminals) connections as once intended. They are jumped together so the thermostat likely wires to the boilers aquastat to TT (thermostat terminals) which interns sends 120volt to the burner and since the primary TT is jump out it comes on.
The EcoBee (and other smart thermostats) need C (Common) to power themselves (thermostat shown has batteries). C (Common) is the return voltage to complete the circuit, although inaccurate you could think of it as a neutral in this case. This is required because the screen, wifi, etc draw too much power for a couple AA batteries.
However I think I see another another conductor behind the wall on the thermostat (a blue wire maybe?). If I'm seeing that correctly that is a good think and you'll need it.
This is what needs to be done.
Turn the power off to the boiler and the part of the AC that houses the low voltage transformer. In a traditional split system (air handler/blower unit inside and AC outside) it is in the AIR HANDLER, not the AC. But make sure BOTH breakers are turned off for the AC to be safe (it will be 2 x 2 Pole breakers). This is important because its really easy to fry a transformer (or blow its fuse if it has one). I would suggest putting the thermostat back on, setting the fan to run and turn off the breakers then verify the fan has shut off. The boiler will be much easier, switch at the top of the basement steps and/or on the side of the boiler.
The thermostat wire from the boiler and thermostat wire from the air handler need splice together or run to the Ecobee's mounting location.
The two wires coming from the boiler (typically red and white are used but color doesn't matter if its an old house it could be brown and brown cloth wires) need to connect to Rh (24volt heat) and W (heat relay) on the EcoBee.
Then you'll need to goto the air handler, find the thermostat wire (small brown wire) remove the cover it runs into. See if there is an extra conductor not being used in the wire (again appears to be behind the thermostat). That unused conductor needs to connect to C on the air handler.
Then at the Ecobee the wires from the air handler will be Red to Rc, White to Y (cool) (typically a yellow conductor is used but there wasnt one there so white was used), Green to G (fan) and that new wire connected to C in the air handler goes to C on the Ecobee.
Here is a schematic I drew up real quick to hopefully help you out.
Excuse my hand writing. That Y wire is jumping over the G wire, not connected together.
Don't concern yourself with the colors of the wires, that is arbitrary and since there isn't a yellow already Y is using a white wire. However do not get them confused, if the boiler is using red and white and you mix up the wires you can possibly damage something and if you damage an aquastat you could be out a few hundred to repair it.
If I was mistaken and there isn't an extra conductor for C (Common) behind the thermostat you will need to run a new wire or sacrifice G (fan) and have no manual fan control (fan will energize with a call for cooling). The included add-a-wire solution (PEK - Power Extender Kit) will NOT work in this circumstance because it uses the wire for the cooling (Y) and heating (W) as a common when they are not being used.
Pulling a new wire from the air handler would be the best way of doing it however to avoid that and sacrifice the G (fan) wire...
Do the same as above except at the air handler instead of connecting the extra conductor (blue?) to C (Common) you will move the green wire from the G (fan) terminal to the C (Common) in the air handler. Then you will connect that green wire to C (Common) at the thermostat. Than take a small piece of thermostat wire (pull a piece off the included PEK unit if you can't spare any in the air handler) and jump G and Y inside of the air handler.
Again excuse my handwriting, I know its embarrassingly bad but it would have taken me much longer to use a program or app.
So any time the Y (cool) terminal is energized the G (fan) terminal will be too. However since there is nothing connected to G (fan) at the thermostat manually setting the fan to run will do nothing. I believe in the setup of the thermostat you can even disable it as an option you access from the thermostat (equipment controls fan).
TL;DR : Call a professional... lol...