Creating address for streets based on direction and hierarchy of streets using ArcMap, QGIS or python?

This is a small example to mimic your case. I have named the layer as 'streets'.

I am using QGIS 3.16.3, which is equiped with overlay_touches() expression in the Field Calculator.

The script is:

if("typeStreet" = 'main street', 
   concat('Alley ', 
          array_to_string(overlay_touches('streets', name, filter:= typeStreet = 'alley')), 
          ', Main street ', 
          "name", 
          ', Highway ',  
          array_to_string(overlay_touches('streets', name, filter:= typeStreet = 'highway'))  
         ) , 
   NULL)

Important part of the script is overlay_touches('streets', name, filter:= typeStreet = 'alley'), which will extract "name" attribute from the 'streets' layer (i.e. own layer) and filter it by the typeStreet. If it is 'alley', it selects the feature fid.1, and 'highway' will be fid.4.

[Logic behind the script]

In this calculation, Main street acts as a hub for the whole alley - main street - highway network. It enables determination of the touching(allay <- main street -> highway) by single command overlay_touches() then use filter to switch between alley/highway. So the calculation starts only if the typeStreet is main street, and ignore other records.

Output will be something like below.