- Firstly, the fact that they were Christians reconquering previously Christian territory would have been enough for them to call it the 'reconquista'. Religious identity was far more important than ethnic or cultural identity in Medieval Europe.

- Secondly, the Spanish monarchy has a direct link to the Visigothic monarchy. The reconquista began with the Kingdom of Asturias, which was founded by a Visigothic nobleman as a rump state of the Visigothic Kingdom of Hispania after the Muslims overran the rest of the country. From the northwest corner of the peninsula it spread south and east, eventually splitting into the kingdoms of Leon and Castille. Castille (to put it very simply) would eventually unify the other Spanish kingdoms and complete the reconquest, forming the Spanish kingdom we know today.

- Thirdly, you're totally misunderstanding the nature of the Visigothic state. Like the Franks in Gaul and the Goths and Lombards in Italy, a class of Germanic aristocrats supported by a relatively small population of their fellow tribesmen ruled over a majority Roman population. When I say Roman, I mean a population that spoke a dialect of late vulgar Latin, was Catholic Christian, and still thought of themselves as ethnic Romans even if they ruled by Visigoths. The Visigothic Kingdom was as much a splinter state of the Roman Empire, like the Gallic Empire or the Palmyrene Empire, as it was a new Germanic kingdom.

As the centuries passed after the fall of the western Roman empire, the hispanic dialects of late vulgar Latin diverged both from Latin and each other, forming Galician (which would become Portuguese), Castillian (modern standard Spanish), and Catalonian (a minority language in modern Spain, but that is an argument that's still going on). This linguistic process was mirrored by an ongoing process of ethnogenisis throughout the early middle ages, whereby the Christians of the Iberian peninsula began forming their own national identities. They had been Romans, but the Roman empire was gone, so they began to evolve identities that took account of both their loyalties to their individual kingdoms and their belonging to the wider culture of Christians of the Iberian peninsula.

Culturally and linguistically, the Visigoths left very little distinct impact. As with the Franks, Goths, Lombards and all the other Germanic barbarians who settled among ethnic Romans, they were assimilated into Romance culture and language.

You may find this earlier thread on the topic interesting. All answers are worth checking out there, and trace the development of the reconquista concept only towards the 13th century, so when most of Muslim states had already been conquered - it was clearly already then a highly ideological concept colored by the Christian states' own religious and political interests. For Castilians this meant "reconquest" in terms of a Christian reconquest, and less so in ethnic terms which would have been difficult after centuries of Muslim rule, cohabitation etc. Although afaik the earliest mentions of Visigothic-Muslim battles (like that of Covadonga) are heavily mythological.

In my answer in that thread I also talk about the term reconquista only becoming more popular in the 1840s. This "meant a change in focus from the „reestablishment“ of the Spanish monarchy and its institutions to a stronger emphasis on the Christian military campaigns – casting these campaigns as as a national founding myth."