It just seems to me that in a time before modern nation-states, calling it a war against Spain is somewhat misleading. If I'm understanding it correctly, most of the armies fighting in the war were mercenaries, with no real nationalist basis for fighting. Likewise the monarch appeared to own the Netherlands mainly as a personal, dynastic possession, with no real ties to Spain.
Background
In 1556 Philip asked the States General for money, which the States General didn't like very much: "Convinced that the Netherlands was paying too much, compared with Spain and Spanish Italy, and that the war was being fought more in the interest of Spain and Italy than the Low Countries, the delegations spent their time airing their grievances." (Israel, p. 136)
In 1559 Philip returned to Spain, to reassert his royal authority as the Spanish monarchy was under threat from France and the Ottomans. In January 1561 Philip had to recall the last Spanish troops from the country, under pressure from the States and nobility. Meanwhile Philip organised, from Spain, a reform of the Catholic Church in the Low Countries. This led to the first revolt (1565). This revolt was suppressed by Margaret of Parma, Philips half-sister and regent of the Low Countries, using local troops.
During the revolt Philip had ordered a Spanish army led by the Duke of Alva to march to the Low Countries. This army arrived after the revolt had ended, but Philip and Alva were determined to use this army to quickly eradicate all subversion and conspiracy, so that the Netherlands would remain secure and mostly Catholic. This had to be done quickly, because the resources would be needed in the war against the Ottomans.
Alva's arrival made things worse instead of better, and his repressive policies forced much of the opposition into exile. But things didn't move quickly enough, and Alva received messages from Philip and his advisors saying that the Duke would have to find regular income from the Netherlands itself to pay for the (still Spanish) army and administration. In May 1568, the King informed Alva: "It is more than necessary to arrange for there to be a fixed, certain and permanent revenue from those provinces for their own maintenance and defence, because clearly the money for that cannot - and will not - continue to come from here." (Parker, p.114)
Alva proposed new taxes to the States General - including the now infamous Tenth Penny, a 10% tax on sales - which the States General refused. Alva put the tax in effect anyway. This made the Tenth Penny "a symbol of unbridled central authority riding roughshod over venerated constitutional procedures, a symbol of government illegality and the ruthless coercion of the towns." (Israel p. 169) The revolt would now begin in earnest (1568/1572).
To answer your question
Philip was lord of each of the Seventeen Provinces in the Low Countries as well as King of Spain (and Naples and Sicily, and England and Ireland, and Duke of Milan). How accurate is it then to say that the Netherlands gained independence from Spain? I'd argue pretty accurate, for these reasons:
In effect, by the time the Dutch revolt really got going, the Low Countries were under occupation by Spanish troops, led by a Spanish Duke, sent there by a Spanish King. Indeed the revolt was justified by saying that Philip failed his lordly duties when he authorised this occupation.
Sources: