The key is in comparing Gallup and Pew's polling data on party identifcation to the findings from exit polls over time. As Cost notes:
Gallup and Pew Consistently Underestimate Republican Identifiers Relative to the Exit Poll. Pew tends to be more pessimistic about the GOP's standing than Gallup, but both always show fewer Republicans than the exit polls. The difference is typically 5 to 7 points. What could account for this? I can think of two explanations.(a) The exit polls are a snapshot of party identification on a single day, while the Gallup and Pew numbers are an average of the whole year. In 2008, both Gallup and Pew showed the GOP at its strongest point shortly before Election Day. Gallup generally showed the GOP stronger in the fall of 2004 than in the Spring or Summer of that year. [Unfortunately, I was unable to locate monthly or quarterly party identification numbers for prior presidential election years.] Why might this be? Some subset of "natural" Republican partisans might only return to their political home when the campaign begins in earnest, around Labor Day. If so, an annual average of party identification - or one that looks at out-years - might systematically underestimate GOP strength relative to where it is on Election Day.
(b) Non-voters are less likely to identify themselves as Republicans than voters, and they are included in the Gallup and Pew numbers. In fact, recent turnout - which is at its highest in some time - is still less than 60% of the voting age population, which means that about 40% of the Gallup and Pew samples in recent years should be non-voters. In a year like 1996, non-voters will constitute more than half of these samples. According to the National Election Study, non-voters are not as inclined to see themselves as Republicans as voters (on a five point partisanship scale: strong Democrat, weak Democrat, Independent, weak Republican, strong Republican). In fact, from 1972 to 2004, the average difference in Republican identification between non-voters and voters was fourteen points. This trend is muted on the Democratic side, as a good portion of non-voters are inclined to see themselves as "weak" Democrats.
I think the take home point from all of this is fairly clear. The Gallup, Pew, and other media pollsters tracking party identification offer data that is of real value - but it has to be interpreted with care. There are big, consistent differences between media polling data on partisanship throughout the year versus the Exit Poll, which is a better metric for partisanship on the day that it matters, Election Day.