Is it theoretically possible for a singular national currency to exist under a free banking system? Free Banking is a proposed monetary system, usually by Austrian Economists, as an alternative to the prevalent Central Banking system. Under this arrangement, commercial banks would issue their own banknotes rather than by a Central Bank. The total quantity of banknotes and interest rates would be dictated by market forces. Free Banking is said to have existed historically in several countries such as the US, Canada, Scotland and China.
The question I'm wanting to ask is concerned mainly with the issue of currencies in this proposed arrangement. Suppose, India tomorrow decides to scrap the RBI (its Central Bank) and move to Free Banking system where banks would issue paper money backed by gold. Would that mean its national currency i.e. the Rupee, would cease to exist because commerical banks would issue their own forms of currencies (Currency X issued by Bank A, Currency Y issued by Bank B and Currency Z by Bank C)? In other words, would multiple currencies (each being issued by a commerical bank) be existing in India then rather than one single national currency i.e. Rupee? Is it possible for India to have a Free Banking arrangement with one singular national currency?
Also, if several competing private currencies are to exist in one country, would each bank be willing to accept the currency of the other? How did historical examples of Free Banking function regarding this issue?
Is it possible for India to have a Free Banking arrangement with one singular national currency?

Under free banking you would only end up with single currency if one bank manages to become a monopoly on issuing the currency. That is possible, although at least historically it did not occurred under free banking systems. Also this would be in essence equivalent to central banking system with only difference that the central bank would not be government institution.
Also, for full disclosure,  I am mainstream economist and not fan of idea of free banking (which is a fringe idea that very few economist support, and it would likely be very inefficient monetary arrangement)  but to my best knowledge of the position of free banking advocates, they advocate free banking because they don't like monopoly on money so ideally the system should be operated with multiple currencies.
However, If the free banking currencies are backed by gold (which was true for most historical free banking systems but does not necessarily need to be part of the system as you can have fiduciary free banking) then even if they technically use different currency they are all in some sort of 'monetary union'. This is because even if they can still vary the number of banknotes redeemable per ounce of gold (and hence the value of individual notes) the exchange rates between the banks is essentially fixed by value of gold which is in practical ways virtually equivalent of being in monetary union (see Eichengreen &amp; Temin, 2010).

Also, if several competing private currencies are to exist in one country, would each bank be willing to accept the currency of the other?

If the free banking banks operate under gold standard then they would have to because in essence what gives value to the money is gold. Lets say hypothetically that under such arrangement bank A would refuse to take money of bank B. Well all you would have to do is just to convert your bank B banknotes to gold and get bank A banknotes for that gold so it is not like bank A can stop you from converting the money.  In addition why would bank that wants to attract customers (and their gold) impose this extra administrative burden on its customers?
If there would some fiduciary free banking then there likely would be floating exchange rates for each banks money and when you would be going to another bank you would just have to pass through exchange rate like when you want to put money in foreign bank. Here I suppose it would be more doable for bank to impose restrictions on other banks money if they would want to - but again question is why would they want to restrict new customers coming to them?

How did historical examples of Free Banking function regarding this issue?

You can find nice accounts of how the system worked in writings of 18th century economists such as for example in Adam Smiths Wealth of Nations. I will only give very brief summary here.
The system developed as an extension of commodity money. Before paper bank notes people used gold/silver (or other metals) coins across Europe. However, travelling with coins is dangerous so people would put them in banks and get 'bank note' of their gold balance that they could get at any other branch of the bank on demand. Since these banknotes were as good as gold/silver coins people started using them instead of the coins.
Under free banking every bank issued their own notes but since the value of note dependent on the gold you can get they could be easily exchanged with each other. If banks A note entitled you to 1 ounce of gold but bank B note entitled you to 2 ounces of gold you would only accept 2 banks A notes for 1 banks B note. Later one as the system developed there were also dedicated exchanges for money issued by various banks.
However, the above is only very rough and crude and simplistic description. The system worked bit differently across different countries since back then there was much less standardization and uniformity when it comes to ways how financial institutions operate (at least relative to present day). For example, the US free banking system was relatively more regulated than lets say the Scottish or Swiss one (see Kenneth (1988) or Kroszner &amp; Randy (1995)).