The answer really depends on how they saw themselves and what time.

Let’s explain the easy part first, the last name: they are known as Balsha(/j) among Albanians, not Balšić. Albanian did not have a standard written language until the 20th century, but those few examples of this time period we have it is enough for us to know that it was very common the spelling of personal names to change depending on which language they were written in, so that alone doesn’t tell us much about ethnicity. This wasn’t an Albanian-only practice either, think about how Polish scientist Mikołaj Kopernik was known as Niklas Koppernigk in German and Nicolaus Copernicus in Latin.

Second part is more difficult to explain but bear with me. Ethnicity used to be a lot more fluid in the past, with people in some areas even identifying with multiple ethnicities at the same time. The strict single-ethnicity thing in the Balkans became more common after the 19th century with the era of nationalism, when it became important to groups to identify themselves on what they valued the most at the time.

Montenegro as a region is actually pretty notorious for this fluid identity, take for example Kuči clan. There is research for them enough to say they were originally an Albanian clan that came to identify as Montenegrins and Serbs over time. Culturally, Montenegro has always been Albanian/Serbian contact zone and as such, people have identified as either Albanian, or Serbian, or Montenegrin whenever they felt like more Albanian, or more Serbian, or more Montenegrin.

Balšić/Balshaj are an example of this too. There were time periods that they identified as Albanians, and even Serbian documents identify them as Arbanasi, and then there are also documents where their Slavic identity is not questioned at all. If we try to understand people of the Middle Ages using ethnic criteria developed after the 19th century, we are completely missing the point.