Does the mental process of assigning subjective qualities to any phenomenon have a name in the suttas?
I think this is what you call Sankara. This is a very complex word with different meaning but you can learn about this in Abhidhamma.

==========

Saṅkhārā is a collective term for the other fifty cetasikas. These fall into four groups:
1) Universal mental factors (sabba citta sādhāranā).
2) Particular mental factors (pakiṇṇakā).
3) Unwholesome mental factors (akusalā).
4) Beautiful mental factors (sobhanā).

http://103.242.110.22/theravadins/English-articles/abhidhamma-in-practice.pdf