They travel into the past, replace the accidentally deceased uprising leader with Sisko and go to the trouble of replacing his identity card at the end on one of the dead bodies. Then Doctor Bashir in the present shows Sisko's photo still went down in history. What purpose does the id swap play in the plot?
The plot states that the ID swap takes place in order to "save the history", in other words: making sure the Bell Riots happen as written down in the history books.
SISKO: For now, but they're the least of our worries. That man who just got killed trying to help us?
(Sisko shows Bashir the food card)
SISKO: That man was Gabriel Bell.
BASHIR: If only I'd had my medkit. If only I'd got him to a hospital I might've been able to save him.
SISKO: You did all you could, Doctor.
BASHIR: But it wasn't enough, was it? A good man died because of us. And what about the hostages? What's going to happen to them?
SISKO: Without Bell, there's a good chance those hostages are going to die.
BASHIR: And if that happens, how is it going to affect the future?
SISKO: We have to save them. Whatever it takes, we have to make sure those hostages survive.
(DS9: Past Tense, Part 1)
The idea is that as little as possible is changed in the past, in order to keep the present intact. Bell's death was seen as an heroic sacrifice:
SISKO: The man they named the Riots after. He is one of the Sanctuary residents who will be guarding the hostages. The government troops will storm this place based on rumours that the hostages have been killed. It turns out that the hostages were never harmed, because of Gabriel Bell. In the end, Bell sacrifices his own life to save them. He'll become a national hero. Outrage over his death, and the death of the other residents, will change public opinion about the Sanctuaries.
So it's not enough to simply save the hostages - to make sure the Bell Riots with Sisko instead of the "real" Bell has the same consequences as the original one, Bell has to die, which is why Sisko has to make sure that there is someone ID'd as Bell among the dead. Can't be much of a hero if you just vanish into thin air!
The article/database entry Bashir finds is likely a compilation from various sources. If you look at the picture, it depicts the outfit Sisko was wearing while in the past. That means it was taken during the riots, perhaps captured from one of the net broadcasts or security cameras. It is not an official government ID photo, like one might have taken at the Department of Motor Vehicles.
In the first part of the episode, when they're initially processed, Sisko is scanned (but not Bashir, at least on camera). There's a device that one might think is a camera, but dialog from the computer terminal a few moments later indicates it was finger prints retinal scan. No pictures appear to have been taken.
The ID swap is to make sure there's a dead body with Gabriel Bell's ID on it, even though it isn't actually the right man. Other than that, it doesn't really matter. When the authorities count up the corpses, they can put "Gabriel Bell" the list. The authorities are not likely to care very much, because the relevance of one terrorist is only important in hindsight.
TLDR: It's a very weak plot point.
A more interesting question, but one that will probably never be answered is why they were still called the Bell Riots, rather than say the Webb Riots?