Civ 1 was clearly designed as a video game from the start. During its first full year of development, it was even real-time (like SimCity and Railroad Tycoon), but Sid Meier wasn’t satisfied with the results, leading to the "valley of despair" he often mentions. The shift to turn-based gameplay came from Empire Deluxe (another video game), which was the main inspiration for Civ 1. The intent was to let players control the pace of the game, not to structure it around multiple players, especially since Civ was always thought primarily as a single-player experience.
The "board game feel" really began with the introduction of 1UPT (one unit per tile) in Civ5, which pushed players toward solving more boardgame-like tactical problems. That direction was further reinforced in Civ6 with the addition of policy cards, adjacencies, and districts. Tiles became increasingly important over time, which is quite distinctive as other historical video game flagships usually went in the other direction.