I wonder if there would be anything alike, with such an edge to it but clean... Cooper seems too round I think?

Any idea when it was put up? If it’s a couple of decades old, it might just be sign painting without a specific font. 

Definitely a Clarendon-inspired piece of type, but agree with Ralf that this particular sample looks hand-painted / lettered.

Similar:

Clarendon Light

Excelsior

News 701 (A and T are too wide, but could be narrowed)

great Excelsior makes sense, as IL MATTINO since whenever it was made surely is a news store, so it might be a hand-made version on that basis (or Clarendon, or Ionic, or News) with sharper serifs... "Six years after the introduction of Ionic No.5, Griffith removed any remaining awkwardness to produce at Linotype Excelsior, which many believe to be the best newspaper text face of all time. It has retained its popularity in Europe, where newspaper columns are wide. Opticon and Paragon are slightly heavier and slightly lighter versions designed for newspapers that deliberately underink to favor halftones, or overink to favor text and headlines."

Grazie!