As there is no mentioning of it in the rules I'm wondering:
Does it make a difference if detect magic is cast from an arcane or divine caster? Thus does detect magic always detect BOTH types of magic or only the same type as the spell is (thus only arcane or divine)
The same question also for identify magic.
Or summarized: Is there any difference between divine / arcane magic in terms of detecting / identifying it (aside from the appropriate knowledge skill)
No, there's no difference. Detect magic would say if there was. Both types can detect the other's magic equally well.
Technically, not the even the different Knowledge skills apply to detecting magic itself. They apply to identifying the results of the two different kinds of magic, though, based on which one creates them most commonly - undead fall into Religion while constructs fall into Arcana. Note that this part isn't straightforward either; both kinds can summon outsiders (Know: Planes) and animals/magical beasts (Nature; which is associated with divine casters specifically through the Druid and Ranger).
For identifying actual spells, either as they are cast or by studying their ongoing effects with detect magic (and identifying magic items with either detect magic or identify), you use the Spellcraft skill. This doesn't change regardless of your or the other caster's traditions. Note that many spells that at first glance clearly belong on one side or the other quickly get mixed up when classes besides wizard and cleric get added to the mix.
Bards' spell lists read more like a cleric's with access to a few good wizard buffs. Cure spells, party buffs, and sonic damage like soundburst....
Domains, familiars, mysteries, and to a lesser extent bloodlines love to grant spells from the "other side". A Fire/Magic cleric has access to common wizard spells, for just a quick example.
