Jake wouldn't be able to replicate latinum for the same reason I can't print off a thousand $100 bills on my laser printer. I don't have the authority to issue legal tender, and if I did, it would be counterfeit currency - useful only until I got caught.
As for not being replicated, we never heard anyone in Trek make the claim. But if latinum isn't replicable, then what is it? Mined? Would the Ferengi allow their currency's value to be tied to the vagaries of the mining industry? Latinum miners would be the most powerful Ferengi in the Alpha Quadrant. They would control the value of the currency, not the market, and not the Ferengi government. But no one ever even hinted that latinum was mined.
So if it isn't mined, then it is manufactured. But why would manufacturing it be possible, but not replicating it? Replication is just rearranging atoms into a pre-programmed pattern. And even if it is manufactured, then wouldn't the latinum manufacturers be more powerful than the Ferengi government?
Whether mined or manufactured, someone is creating currency out of "thin air", just as a replicator would. I don't see why latinum would hold value better than anything else just for not being replicable. Latinum's only use is as currency; it's never suggested that it can be used in anything else. Sure, some decorative items, but no industrial use for latinum is ever described. At least gold is used for industrial applications in addition to looking pretty.
So what is latinum? It's fiat currency. It has value because everyone agrees that it has value.
Again, Morn was chased for liquid latinum in his stomach. And you can't replicate bills because they have serial numbers and other safety precautions. Nobody chases you for the paper or the ink.