Does this refer to Alfred's father Aethelwulf? If so, why is he described as "King of the Saxons" as opposed to "King of the West Saxons" or "King of Wessex"?
Or was this "King of the Saxons" a different person altogether?
Was this a title bestowed upon this Aethelwulf to fend of the Norsemen, or is Asser exaggerating the power of Alfred's father to please the West Saxons?
Yes, it is his father.
If you read the ASC s.a. 827, Ecgbeart is classified as a bretwalda "ruler of Britain" (or as Kirby has it more accurately, brytenwealda - a wide ruler of England) and the ASC is trying to show that they ruled all of England. The term doesn't appear before the 9th century, so this is an attempt to show that this particular lineage ruled everything. We can tell this because the chronicler attempts to link him with Bede's list of 7 kings who ruled all the Anglo-Saxon kingdoms south of the Humber.
Ecgbeart had to wrangle with the church at Canterbury (and others) to establish Aethelwulf as his heir so his position wasn't as strong as the ASC described (at least initially), but it was a substantial achievement, and one with Aelfred inherited.