There were several sources through which 16th century writers, poets and political philosophers of the Polish szlachta came to contact with the term "Sarmatians". Perhaps the most striking source of information was the Geography of Ptolemy, an atlas of the world composed during the late Roman period, which was translated to Latin in 1406 and disseminated in late medieval Europe, including in Poland, where maps of the region relevant to them, Eastern Europe, were marked as Sarmatia Europea and Sarmatia Asiatica. To Greco-Roman cartographers, Sarmatia, in spite of its connection to the Sarmatian peoples, was also a wide geographic term encompassing the parts of Eastern Europe to the east of Germania, in which Sarmatians were only one of many tribes - but this original meaning was lost over the centuries, and gradually, in Poland, this label was reshaped to an imagined Sarmatian empire, ruling over much of Eastern Europe and lording over the local Slavs - much like Polish szlachta, in the 16th and 17th centuries, saw themselves as the lords of a great Eastern European empire. Equally important were the works of Herodotus, which were known in Poland at the time much like they had been known in the rest of Europe - they returned to Latin European culture through the influx of Greek scholars to Italy carrying and translating ancient works which had been preserved in the Byzantine cultural corpus. They, as well as other authors such as Aristotle, were condensed into the Tractatus de duabus Sarmatiis (Tractate on the Two Sarmatias) by Maciej Miechowita in 1517, in which the idea which would later develop into Sarmatism was first established. Miechowita's Tractatus was also one of the first in-depth ethnographic and geographic works on Eastern Europe and its prestige in this regard meant that it was rarely questioned in the following centuries - and so, the theory on the Sarmatian origin of the Polish gentry was unchallenged as well.

To address the specific point you made in the description, clarifying over whether it would have been understood that the Sarmatians spoke an Iranian language, the knowledge and understanding of linguistic relationships in general was still extremely vague and far off from the classifications we have today. Eastern Europe was no different in this. To make an example of how much intellectuals of the time knew about linguistic relationships, around the same time as Sarmatist ideas first started to develop in Poland, the diplomat Michalo Lituanus wrote a treatise dated to roughly 1550 and titled "De moribus tartarorum, lituanorum et moscorum" ("On the Customs of Tatars, Lithuanians and Muscovites"). In this treatise, among other things, Lituanus (generally believed to be a Protestant noble of Lithuanian birth) describes the similarities between Lithuanian and Latin, pointing out 74 similar sounding words and stating that this is proof... of a mythical origin story of the Lithuanian people as refugees from the Roman Empire who settled in modern day Lithuania. In fact, he even follows this statement by calling for Latin to become the official language of the Grand Duchy of Lithuania, as this would supposedly restore the "native language" of the Lithuanian people!

Now, five centuries later, we know that these similarities exist because Lithuanian and Latin are both conservative Indo-European languages, but at the time, this genetic similarity was completely obscure and so was the existance of language groups making up the IE language group. And for a language like Sarmatian, which did not survive to the modern day or even the 16th century, the chance that anyone would have made the connection between Sarmatian languages and the surviving Iranian languages is effectively nonexistent.

Somewhat ironically, even though Sarmatism has been dead ever since the Early Modern Period, today modern linguistics find borrowings from Indo-Iranian languages in Proto-Slavic (such as *bogъ "god" and *rajь "paradise") which can only really be explained through borrowings from Scythian or Sarmatian, indicating that they may have played a notable role in very early Slavic history.