How to fix this unusual looking valve stem to put air in the tire?
Your valve looks like a regular Presta aka 'french valve' to me. It's just a bit on the short side for your current rim.

I would recommend using a Presta valve extender, it could replace the small nut that is currently located on the valve stem assuming the valve hole in the rim wall isn't drilled out for a bigger size valve (the bottom of the valve extender will then push against the rim thus taking the function of the small nut that is currently installed). You could also just remove the small nut (once again assuming rim wasn't drilled for bigger size  valve stem) so you have a bit more room to install a pump. just make sure to depress the tire at the location of the valve as to prevent it from being pushed down when installing the pump.

Presta valve extender(example):



You could also try using a adapter to go from Presta to Dunlop or Presta to Shrader, if you remove the small (currently installed ) nut it looks like you should have enough thread sticking out on your current valve stem in order to install such an adapter.



Source

bonus info:
https://www.bicycling.com/repair/a20048610/the-difference-between-a-schrader-valve-and-a-presta/