Im not really familiar with boilers, its designs and the applicable design codes. However, the PED treats boilers differently from pressure vessels (and thus heat exchangers), when it comes to classification. The idea behind this different classification is (I believe) based on the old Stoomwezen rules, the first Dutch set of rules pertaining to design of pressure equipment. The rules where set up when the industry started using steam as a means for power in the late 19th, early 20th century.
I dont have detailed information at hand why this classification was treated differently, but from what I can (best) guess, the reason may be that the risks involved in boiler designs are different from ordinary pressure vessels. The PED looks at pressure equipment from a risk point of view. The higher the risks (usualy due to volume and pressure), the higher the assessment class. Boilers have a risk of overheating the tubes by the hot flue gases, when there's no or too little product feed. This may push the surface temperature beyond the design temperature or acceptable skin temeprature for the material involved. Such risks are normally not applicable to pressure vessels.
Sure, the rules in ASME are different from PED: one being a design code, the other being a directive with no design rules (at all). Just my 2 cents, not sure if it contriibutes anything to this discussion.