A theory shared to the "Sopranos" subreddit by u/anubis2051 in 2014, which has also been shared among fans of both shows since "Lilyhammer" premiered, proposes that Frank Tagliano is a functionally Silvio dreaming. Proponents of this possibility cite both precedent in "The Sopranos" and what can be interpreted as confirmation in "Lilyhammer."
In a post about the connection between the two shows shared to the Fan Theories subreddit in 2014, u/gunslingrburrito likens Frank to Kevin Finnerty, a character Tony imagines himself to be in an extended coma dream sequence following a gunshot wound in "The Sopranos" Season 6. In the "Sopranos" series finale, Silvio similarly ends up in a coma as a result of multiple gunshot wounds. It's at least plausible, then, that he might dream up an alternate persona like Tony did when imagining himself to be Kevin Finnerty.
Furthermore, in their post to the "Sopranos" subreddit, u/anubis2051 mentions that in "Lilyhammer" Season 3, an old man says the words "it's all a dream Silvio." While many commenters in the post's reply section believe this is simply a reference on the writers' part to "The Sopranos," assuming the coma fan theory to be true, this could be Silvio's subconscious acknowledging the nature of his extended Frank Tagliano dream.
For what it's worth, Steve Van Zandt discussed the theory in an interview with Maxim magazine. "Those of us who've done more research into the mob know how different the two characters are, but if someone thinks it's that, that's fine – I don't mind that at all. Whatever allows people to be comfortable watching the show is great," he said, authoritatively confirming the theory as non-canonical while still allowing room for its proponents to have fun with it.