Does this thermostat control heating and cooling?

It controls everything! Air condition, Heating, Eheat and Vent. I believe it is the Aube by Honeywell TH135-01

Actually it is not the TH135-01 since this one only does heating. But it looks exactly the same.

Here are some pics

The Honeywell TH135-01 is a heating only thermostat for hydronic systems. If this a 24V thermostat, then you have many programmable thermostat choices. You would need a third wire connected to the C terminal, or a ground connection through the house wiring if you want to install a WiFi thermostat.

I just opened and look at the model number of my thermostat TH146-N-2H1C/U. Do you know if there is an equivalent one that is programmable and that I can simply swap without changing that black box in the basement?

You have a special thermostat which operates with a controller. It uses two wires to control heating and cooling. You can't replace it with a standard 24V programmable thermostat without adding more wires.

The version you have is non-programmable. There is a TH146-P-2H1C model which is the programmable ("N" versus "P" in the model number). However this comes with the control model which you already have. That's spending a lot of money for something you don't need.

I suggest you call Honeywell and ask them if there is a programmable console (thermostat) you can buy that is compatible with your controller. I looked for one but I was not able to find it.

Same thermostat does not