Well, it's FALSE identity. Of course you're not really a duke, or an acolyte, or a researcher, or a folk hero. But you don't need to be. You just need your targets to believe you are. Once you've bluffed your way into the temple, you might persuade a religious order to shelter you as one of their own. Or maybe you can fool the innkeeper into thinking you're a folk-hero and con him into giving you free beer.
So yes, you can receive benefits similar to the "Noble Background". However, there are a few limitations:
It's not an automatic success. You have to convince your audience that you are, indeed, the Great Duke Roscoe. It'll be relatively easy to convince a passing traveler of your identity. It'll be much more difficult to persuade noble well-versed in the politics of the city.
It's not a permanent success. Somebody can realize you're a fraud and stop giving you any benefit (or they might actively seek revenge).
It can backfire. People don't just give you stuff for free. They want something in return.
- "How convenient that an able-bodied acolyte appeared just when we needed a quest done! It must be a sign from God!"
- "You're the hero who stood up to a tyrant and led a people to freedom? Amazing! I've got just the tyrant for a righteous hero like yourself to overthrow."
It can collapse. Maybe you've been running this scam for years, and people have been warned about "The Duke Roscoe" as a textbook example of a scam. Or somebody's put him on a hit list. Or maybe you've been summoned to appear before the prince for a formal investigation of your claim to royalty. You might need to ditch that identity permanently.
In a comment, you asked about acquaintances. Yes, you have acquaintances who will say you're a duke. But remember, you don't have the wealth, the land, or the political heft to back up your claims. You can have two types of acquaintances: co-conspirators and gulls. The co-conspirators know you're a fraud and expect a share of the profit. The gulls do not.
You'll need to coordinate with your co-conspirators to build a consistent identity. You don't want them to slip and call you "m'lord Baron Schemer" when you're playing the Duke Roscoe.
For the gulls, you'll have to maintain their belief, or else you'll lose them. If they discover you've been conning them, they may try to get revenge. You also must not gull the wrong people. You do not want the mafia to hear that there's a vulnerable noble just begging to be kidnapped for ransom, and you do not want the nobles to stomp out a potential competitor who's been hanging around the slums.
If posing as a fallen noble
If you're playing a fallen noble, then you'll lose some advantage in one of the four areas above, or in the nature of the benefit. Fallen nobles might be scorned by regular nobility, for instance. Or it'll be tougher to convince people. "Oh, sure you're a fallen noble, and my uncle's a Sahuagin".
Disadvantages during the scam
Normally, I'm going to fall for your scam because the greedy part of my brain overpowers my skepticism. Once I realize that you don't actually have anything, my greed's going to back off. Now, greed and skepticism both come to the forefront of my mind and demand: Am I actually going to get paid or not? I'm starting to doubt whether you can deliver, even if I'm 100% convinced you are a bona fide duke.
In the long term, you're sitting on a time bomb. The people you do manage to con are going to come calling and asking what your status is on reclaiming the throne. Eventually, you'll gain the reputation of that one duke who lost his throne, promises everyone riches, but never delivers. That can still be your false identity, but don't expect to get much of a bonus from it.
Exception: Long-Term Scam
If your scam depends on a consistent story, playing a fallen noble is easier, since your identity will survive some scrutiny. This is usually more in line with big target with a long setup: a mass of people, a prince, or maybe a wealthy guild. Once the setup is done (which could take months), you pull the string and vanish with a ton of money. That's hard to do without making a campaign out of it; check with your DM (and the rest of your party) beforehand to make sure that this is something they want to do.
I'd recommend you try fleshing it out some more. What does a typical encounter look like? Who are your targets, and what are you trying to gain? What kind of reputation will you accumulate as time passes?