Is Kingmakers single-player?
Yes. The campaign is built to be played alone and works without a connection, and up to 4 players can also run it together. The developers describe it as a single-player campaign that is also playable in co-op, not the other way round. What alone does not mean is fighting alone: you still command an army, and later missions are not meant to be soloed.
Solo, offline, with a tutorial and no forced matchmaking, by the developers’ own answers. What would change this: a developer post limiting solo play, or the store pages dropping the Single-player label.

What the developers say
Solo, offline
“Yes, you will not require an online connection to play solo.”
Official FAQ. From the Kingmakers Official FAQ (Steam discussions)
Designed for one
“We’ve designed Kingmakers to be a great single-player experience.”
Redemption Road. Said in the tinyBuildGAMES video: Kingmakers - Gameplay Deep Dive
Never changed
An account carrying the developer badge wrote that the design has not changed since the first announcement, and that “It has always been a singleplayer campaign that’s also playable in coop.”
How solo play works, as far as it has been said
The private-lobby and tutorial answers are quoted in full on Kingmakers co-op and multiplayer and the Kingmakers Q&A. The director’s wording on lobbies was that “You can play in singleplayer. You can have a private lobby and invite just your friends.”

What alone does not mean
Playing solo removes other players, not the enemy armies. The Deep Dive puts it plainly: “Even with an army at your side, you’re still heavily outnumbered.” You recruit and command that army from the strategy view, covered on Kingmakers units and officers and Kingmakers base building.
The director is more specific about how far one player can get without it:
“A man by himself with an assault rifle loses against 30 archers.”
“Maybe not at the very start, but on harder difficulties and later missions, you really can’t solo everything even with air support and turrets and landmines and drones.”
So a solo campaign is a commander’s campaign. The difficulty side of that is on how hard Kingmakers is, and how a co-op group divides the same battle, each player with an army of their own, is on Kingmakers co-op and multiplayer.
Where the store pages and the shows stand
Steam files the game under Single-player alongside Multi-player, Co-op and Online Co-op, and the Epic Games Store lists Co-op and Single Player as its features; the full list is on Kingmakers crossplay. The one build the public has played, at BiliBili World, was single-player, as written up on the Kingmakers BiliBili World demo, while the planned demo is described as fully multiplayer; see the Kingmakers demo. What the first Early Access release includes is on what Kingmakers Early Access includes.