Up until just a few years ago, the neutral was not required at a switch location.
A standard switch loop consisted of the hot feed to the switch, the switched return leg, and a ground wire. With non-metallic cable you have a black wire, a white wire, and a bare wire for ground. The National Electrical Code requires that the feed to the switch be the white wire re-identified as any other color but white or gray. So, with cable, you would then have a black wire, another black wire (a white wire with black tape on it since that is what the electrician carries all the time), and the bare ground wire.
If you had conduit running to the switch box then they would just pull two black wires for the switch loop, and a green for ground.
The Code now requires a neutral at most switch locations which is usually a white wire but could also be gray. This was not the case when your wiring was installed and is still not required if the installation uses conduit.
So, your switch box most likely had one black for a hot feed, another black for return to a load (fan, light etc.), and a red for return to aother load. And of course the green ground wire.
It will work the way you have it but one of the wires is extra.
Good luck!
Lets start again:
Unless you have turned off power at the breaker, keep both switches off once you have a fan in place without blades. Powering a fan without blades on can damage the fan.
You have determined the black wire on the left is the line hot, and it seems this must be connected to the lone white (repurposed as hot) next to it. This carries the line hot to both switches. (That white does connect to both switches, right?) In the same cable as this white a black comes from one switch and a red from the other comes from the other. Pick one (say the black) and connect it to the black and the blue leads of the indoor fan. A switch now controls the hot to the indoor fan.
Now connect the white lead of the fan to the two connected white wires (these are neutral). Connect the bare copper to the green w/ yellow stripe. At this point the fan would work but don't try it without the blades attached (could damage the fan motor) and don't put them on yet because you still have more wires to connect for the outside fan and the blades would be in the way.
If you want you could put the blades on and test the indoor fan, then take off one blade to give yourself room to make the further connections. But at this point I think there are only two wires not not connected the red and a black going to the outside fan. Connect these two wires to take switch hot to the outside fan.
Both fans should now work.
Best Answer
The answer is yes you can run your 2 hots the same color and they can be any color except white, gray, or green. If your circuit is only 220. (240 in modern terms) you will also need a ground, the ground can be metallic conduit or if using pvc conduit a green is normally used. If you want to run anything other than 240v you will also need a neutral to set up a breaker panel. If this is a detached shed a ground rod will also be needed. With more info we can help you size the wires & conduit based on the load and distance but more info will be needed to do that but Yes you can use 2 black insulated conductors.