I think the convention of tuning your starting note to a piano key frequency, and then referring to that frequency using the piano key name is a recent phenomenon in Indian music. It is something musicians are doing these days because they find it convenient when communicating about the music or collaborating with other musicians in a global or digital context. But the music itself does not depend on either the terminology or a specific reference pitch. You could just as easily use some other terminology or reference pitch, and the music would not be affected.
Again, note that it is only the root note (sa) that uses the piano-key reference pitch. The pitches of the other notes are relative to "sa" and depend on the raga. A sitar, for instance, has to be retuned each time a new raga is played on it. So, tuning in Indian music is actually quite a sophisticated science and very deeply rooted in an understanding of consonance between different sets of notes in different ragas.
This has been the case since ancient times in India. Bharata's Natya Shastra (written sometime between 200 BCE and 200 CE) describes a tuning experiment he (Bharata) conducted using two vinas. He used one vina as the control and re-tuned the second vina to get a slightly different tuning (between the two tunings – called shadja graama and madhyama graama – several jaatis (modes) could be played harmoniously on the vina). Bharata's experiment was intended to prove the relative pitch locations of the 22 shrutis in an octave.
Tuning to harmonize multiple instruments would also have been an essential aspect of ancient Indian music culture in general, as orchestras (called "kutapa" or कुतप) involving multiple classes of instruments were very common in Bharata's time. But musicians who performed together would naturally have been present in the same location at the same time and could tune their instruments to match each other without the need for an absolute reference pitch/frequency. (Having said that, there are any number of ways to get a reference pitch - temple bells, for instance).
As for the convention regarding the naming and pitches of the different swaras, one convention is that pitches are relative, not absolute. As you yourself said, it is the intervals that matter in Indian music, not the specific frequencies of notes.
However, the sequence of sa re ga ma pa dha ni (full names: shadja, rishabha, gaandhaara, madhyama, panchama, dhaivata, nishaada) has existed for as long as we know. Ancient texts even predating the Natya Shastra mention these note names in that same sequence. But originally, "sa" was not fixed as the permanent root note of the octave. Modes (jaati) were derived by modulating, i.e., changing the root note.
The history of the formation of the octave in Indian music is a vast topic that requires its own very detailed answer.
