There is a strong association between eczema and childhood asthma. This association has been documented in pre-adolescence, adolescence and adult life, into middle age. If one or both parents have a history of eczema and/or asthma, the incidence of combined asthma and eczema is even higher. That being the case, does not necessarily mean that your son has asthma; just that his odds of having it are higher than in children similar to him, but without eczema.
However having a “rattling in his breathing for 4 weeks” is not a basis for making the diagnosis of asthma. Four weeks is more consistent with an acute, usually viral, respiratory infection. That his Pediatrician has observed it to be “coming from high up”, further weakens the likelihood that this is asthma. Trust his Pediatrician and if the rattle persists, for a couple months or more, you might want to have your son’s lungs reevaluated.
Good luck
Does he have reflux? I would find a specialist, possibly start with an allergist or ENT or at minimum see another pediatrician. It is not right that he has had a permanent cold for that long and has rattling breathing even if it is high up. High up usually means croup but it wouldn't last so long. Be proactive find another doctor.