Current, with 45 employees and 350,000 accounts, is tiny compared with the grand vision Facebook promoted in interviews this week. But Sopp said Current was launched with a similar belief: The existing financial system isn't serving large swaths of people. The start-up first targeted teens with a banking product and this year widened its appeal by offered no-fee checking accounts geared toward gig economy workers.

"This is a funny way to try and create trust in a new global financial system – by ripping off another fintech firm," Sopp said in a phone interview. "Facebook has all the money and resources in the world. If they truly wanted to make banking more inclusive and fair, they should've come up with their own ideas and branding, like we have."

Anthony Harrison, a spokesman for Facebook, declined to comment for this article. Character did not return calls and messages for comment.

Stuart Sopp, a Wall Street trader turned start-up CEO, told CNBC that he used a San Francisco-based design firm named Character to create his start-up's logo in 2016. That firm also worked on the secretive crypto project that Facebook unveiled this week, according to a LinkedIn post from Ben Pham, creative director at Character.

The founder of online bank Current claims Facebook copied his company's logo for the social network's bid to reinvent the global financial system.

Sopp spent months working with Character to come up with his firm's logo, he said. It's meant to capture the importance of movement – of both money and people – by representing a wave. Current applied for trademarks in 2016, which are pending, Sopp said.

"We put six months of hard work into this with that design firm, which they basically reused for Facebook without changing much," Sopp said. "Facebook is a big company that should have done their due diligence on this."

Current engaged a law firm, Goodwin Procter, to determine if it has a trademark or patent infringement case, Sopp said. The firm did not return a call for comment.

link to tweet

He was attending a fintech conference held by Fortune on Wednesday when Sopp got screenshots of the logo for Calibra, the subsidiary Facebook created to house its financial efforts. He was shocked at the similarities and initially thought he was being pranked, Sopp said.

Calibra will offer a digital wallet to store and spend Libra, the new cryptocurrency Facebook hopes will someday be the coin of an alternate financial system. If it takes off after its introduction next year, Calibra could offer other financial services to users, like loans or investments.

To succeed, Facebook has to convince regulators and elected officials around the world that the benefits of allowing the rise of a new global financial system powered by cryptocurrency outweighs the risks.

But if Facebook cut corners on something as basic as the branding of its nascent efforts, this dispute could give ammunition to critics who warn that the social media giant can't be trusted with more of the public's data. Politicians and central bankers in the U.S. and Europe have already vowed to investigate the company's plans.

Sopp said he sent a direct message on Twitter last night to Cameron and Tyler Winklevoss, the Harvard twins who famously claimed that Facebook founder Mark Zuckerberg stole their idea for a social network.

The message: "Now I know how you guys felt," Sopp said.

— With reporting by CNBC's Salvador Rodriguez.