We know very little of the political system of Panem, as the Capitol tends to keep its citizens and the districts ignorant. And since everything is told in Katniss point of view, that means we learn very little.
We do know that Panem is set in a post apocalyptic North America, and can simply conclude that certain terms were kept. They all speak English with peppering of Latin names and terms. Panem grew out of a war and environmentally ravaged United States. It would be a slow transition of terms, so President would likely stick.
It is possible that Snow was elected, based on the secrets Finnick learned when Snow was pimping him out. From Mockingjay chapter 12, very close to what is shown in Mockingjay Part 1:
"And now, on to our good President Coriolanus Snow,” says Finnick. “Such a young man when he rose to power. Such a clever one to keep it. How, you must ask yourself, did he do it? One word. That‘s all you really need to know. Poison.” Finnick goes back to Snow‘s political ascension, which I know nothing of, and works his way up to the present, pointing out case after case of the mysterious deaths of Snow‘s adversaries or, even worse, his allies who had the potential to become threats.
As he says, Snow rose to power through political alliances and murdering his opposition. The position of President could be a life Appointment since Before the Dark Days i.e. the first rebellion, or Snow convinced them to give him that power ala Star Wars Emperor Palpatine in Clone Wars.
As for the voting, obviously the Districts, which after the Dark Days are no longer given various freedoms, that leaves Capitol Citizens as the only Voters. We don't know if District citizens/residents could vote Before the Dark Days. The oldest characters we see are Mags, born 5 years before the Treaty of Treason, and Snow, born that year, 75 years before current. So any first hand account is hard to find. In comparison, modern day United States does not allow citizens living in its Territories to vote for President or Congress. Puerto Rico is an example of that. District of Columbia citizens can vote for president but not for Congress. So there is precedence for how the Capitol treats the 12 Districts. From the first book:
[The Mayor] tells of the history of Panem, the country that rose up out of the ashes that was once called North America. He lists the disasters, the droughts, the storms, the fires, the encroaching seas that swallowed up so much of the land, the brutal war for what little sustenance remained. the result was Panem, a shining Capitol ringed by thirteen districts, which brought peace and prosperity to its citizens. Then came the Dark Days, the uprising of the districts against the Capitol. Twelve were defeated, the thirteenth obliterated. The Treaty of Treason gaves us the new laws to guarantee peace and, as our yearly reminder that the Dark Days must never be repeated, it gave us the Hunger Games.
In Contrast, we also don't know much about District 13's President Coin, if she was democratically elected or not. We do know that after her and Snow's deaths, Commander Paylor was elected as the new President of Panem in an emergency election, so Panem/District citizens clearly understand the concept of a republic.
Tonight, the 12 district leaders will call for a free election. There's no doubt that Paylor will carry, she's become the voice of reason.
-[Haymitch reading Plutarch's letter, due to Hoffman's passing]
Of course, Plutarch is the source of this concept. From the book Mockingjay:
If we win, who would be in charge of the government?” Gale asks.
“Everyone,” Plutarch tells him. “We‘re going to form a republic where the people of each district and the Capitol can elect their own representatives to be their voice in a centralized government. Don‘t look so suspicious; it‘s worked before.”
“In books,” Haymitch mutters.
“In history books,” says Plutarch. “And if our ancestors could do it, then we can, too.”
[Katniss, Internal] Frankly, our ancestors don‘t seem much to brag about. I mean, look at the state they left us in, with the wars and the broken planet. Clearly, they didn‘t care about what would happen to the people who came after them. But this republic idea sounds like an improvement over our current government.
In Mockingjay Part 2, Coin "promises" a free election, when when the time is right. Which we and Katniss know is B.S., leading to Katniss' actions at Snow's execution ceremony.
