Is the value of nano arbitrary? There is no work required to produce or obtain it.

I agree with Colin that the labor theory has no supporting evidence.

Setting the value of cryptos on how much work or energy they required (typically to be mined) is a non-sense. It is like saying that a painting has more value because the artist spent many hours painting it.

Like user fatalglory, I believe that Nano's value should be see as the value of the network, i.e. the value of the service it provides.
Additionally to PayPal, I would suggest Visa and Mastercard as a comparison.

The total supply of NANO is around 133 Mio. Currently the NANO is traded around 0.73 USD.
So, the total value of the NANO network is currently priced a little below 100 Mio USD (133 Mio * 0.73$). Nowadays, this is certainly a lot.

But one can see things differently for the long term: if most merchants accept NANO, they will spare in fees and (bad) exchange rates. The value of the Nano network can be see as a consensus around how much the service of feeless transactions is worth as compared to the fees that merchants and exchange rates that their customers have to pay for nowadays popular payment services (PayPal, Visa, Mastercard, ...)

Cryptocurrencies can be forked (and are forked): there are many Bitcoin-based forks, and for Nano is the Banano fork.
Most cryptos are less prone to hyperinflation than FIAT currencies because their total supply is fixed. Hyperinflation in the number of cryptos offered is however possible, because cryptos can be forked. Today we can observe a bubble in the number of crypto currencies available. The ones that will survive are not necessarily the best ones from a technological and ecological point of view.

Few challengers have the potential to become global leaders. Few cryptos will probably dominate the market. One can compare this similarly to credit card companies (Visa, MasterCard, American Express, MIR, ...)