Is there a qualitative difference between the rent derived from a supply fixed by nature and single supplier required to supply a fixed quantity?

A monopolist prices by setting marginal revenue equal to marginal cost. The marginal revenue depends on demand.

If you have a fixed supply in the sense that the quantity offered is always some $qin mathbb R$ independent of the price, then you define a market equilibrium such that the demanded quantity at the equilibrium price must be equal to $q$. As a dictator you may also impose that the entire quantity is traded at price zero and then somehow ration the excess demand, but then --some might argue that-- a secondary market would form such that the consumers with the highest willingness-to-pay end up with the good anyway.

If you have a monopolist who can produce up to $q$ units at cost zero (put differently, this monopolist already has $q$ goods and production is impossible), this monopolist would either trade the entire capacity if at $q$ marginal revenue > 0 OR trade up to $q'<q$ units where marginal revenue is zero at quantity $q'$.

Answer to Original Question Assuming Limited Supply

In my opinion equating the two is fallacious from outset. The reason for this is that you can have cases where you have both monopoly and limited supply, only limited supply or only monopoly or neither. In addition, these have different implication for firm firm behavior in general depending on market structure and form of competition.

For example, consider a trivial example of competition under Bertrand duopoly with firm A and B with demand of individual firm assumed to be $D_i = Q/2=100-p$ (where $Q=q_A+q_B$) if prices are $p_A = p_B$ (e.g. if firms charge exactly the same price they each get half of the market demand). If $p_A>p_B$ firm A only gets residual demand after demand of firm B is satisfied (that is in this case $D_i = Q-q_j =100-p_i$ , and if $p_A<p_B$. Moreover, we will assume that marginal costs are equal to $c_i=10$ for both firms

Let us start by assuming that both firm A and B production cannot exceed 100 i.e.: $q_A,q_B leq 100$. In this case clearly the Nash Equilibrium (NE) will be given at a point where $p_A=p_B = c$. The reason for this is that in this case the fixed supply is large enough to satisfy the demand so if any firm would dare to raise prices above $p=c$ it would lost all the demand to the another firm. Also in this case firms will have zero profit, and quantity sold in the market will be $90$.

However, consider monopoly in such situation. Again let us assume demand is the same $Q=100-p$ (here naturally whole $Q$ goes to the monopolist) and let us again assume $c=10$. Furthermore, let us again assumed supply cannot be higher than $100$ so $qleq 100$ In that case the profit would be given by:

$$pi = (100-Q)Q - 10Q $$

and it is trivial to see that optimal profit maximizing quantity is $ Q^* = 45$ and consequently $p^*= 55$ and $pi^*= 2025 $.

In both cases we have some restriction on supply and in one case we have no profit and $Q=90$ and in second case we have quite a large monopoly profit and $Q=45$.

Now, of course in the above we get this large contrast also because I assumed that the restricted supply is still larger than the maximum demand. However, even if we would make the supply restriction more strict we would get whole range of quantities at which the outcomes between monopoly and Bertrand competition would not be the same. Eventually as we would start restricting quantity further there would be a special case where Bertrand Duopoly and monopoly would have exactly the same outcome. Hence, I won't deny there are special cases where in terms of outcome monopoly and restriction of supply will get you the same result.

But those are special cases not general ones. Generally you cannot equate restriction/limit on supply with monopoly. They even can exist jointly and independently of each other. I mean there are special cases where monopoly charges the same price as perfectly competitive firm (e.g. perfectly elastic demand) but it would be absolutely inappropriate to conclude that there is no difference between monopoly and perfect competition.

Answer to Edit

If you assume that quantity is fixed at some $bar{q}$ and it cannot change then it is trivial to prove that generally $q$ supplied by monopoly wont be equal to the case of fixed supply.

In the monopoly example above we found that monopolist would supply exactly $q^*=45$ - no more no less. The quantity that monopolist chooses is not random - it is literally engineered to maximize the monopolist profit and to get as much profit as possible.

However, $bar{q}$ fixed supply that must be brought to market will almost always give you different outcome as in this case there is no reason to assume nature chose $bar{q}$ to maximize anyone's profit. For example, if $bar{q}=10$ then price on the market in the example above would be $p=90$. Moreover, if there would be multiple firms lets say 10 firms all offering 1 of those $q=10$ products profit would be just $90$ per firm, if $bar{q} =60$ price would be $P=40$ and again we assume there is 10 firms individual profits would be $240$.

Literally only in most special case with all fixed supply provided by singe firm and fixed supply happening to be exactly $q=45$ - which is astronomically unlikely to happen at random would you have a special case where market outcomes are identical between restricted supply and monopoly case.