How is it that Spain and Portugal have such distinct histories and languages even though they have shared the same peninsula and the territory for so long?

Portugal actually predates Spain as we know it by several centuries. The Kingdom of Portugal split from the Kingdom of León in the 12th century (with Afonso I becoming the first King of Portugal in 1139), and upon its independence it became one of the several different independent powers on the Iberian Peninsula, some of which were Christian, some Muslim. Traditionally, the unification of Spain is dated to 1469, when King Ferdinand II of Aragon married Queen Isabella I of Castile and León, thereby uniting what by then had grown to be two of the main powers on the peninsula - the Kingdoms of Castile and Aragon. However, Spain as an actual nation was not formed by this act, and the two "Catholic monarchs" ruled over a composite monarchy that really was two different kingdoms. The Emirate of Granada - the last remaining Muslim power in Ibera - was taken by Castile and Aragon in 1492 and the Kingdom of Navarre was conquered by the 1510s. With that, the Iberian Peninsula, save for Portugal, was effectively united under one crown, though the various kingdoms that made it up would continue to exist as political entities for quite some time.

I bring this up to illustrate the fact that Spain is itself a highly diverse polity which was made up of several different entities that all had their own distinct histories, cultures, and even dialects or languages. If you were to go back to 1400, the idea that Portugal was somehow significantly more different from Castile than Castile was from Aragon (let alone Navarre or Muslim Granada) would have seemed a bit off. And keep in mind that Castile and Aragon themselves had taken over or merged with other entities prior to their union in the 1400s. Portugal was one distinct region of Iberia among many distinct regions of Iberia. In the several centuries that Spain has been effectively unified, some form of national identity has certainly arisen, which is why we can easily today point towards the Portuguese as being different from the Spanish, but even today there are certainly strong regional differences between the peoples of Spain (to the point where some of them don't actually want to be a part of Spain). Portugal just managed to remain a separate kingdom, unlike those other various regions.

Following a succession struggle, Portugal actually did enter into a dynastic union with Spain in 1580, with Philip II of Spain being crowned Philip I of Portugal. Philip II and Philip III (I'm using their Spanish titles here) generally respected and admired the Portuguese, and Portugal retained its administrational independence under the united dynasty. However, Philip IV became very unpopular with the Portuguese nobility and merchants due to heavy taxation and his attempt to make Portugal a royal province (at the expense of the power of the Portuguese aristocracy). Portugal's nobels rebelled, crowned John, Duke of Braganza as (who was the grandson Catherine, Duchess of Braganza, who was one of the claimants back in that original succession struggle) King John IV of Portugal, and succeeded in reasserting Portuguese Independence. Had the war of gone the other way and Philip IV managed to carry out his designs, perhaps Portugal would be looked at as simply the last Iberian kingdom to be integrated into Spain.