The Highland League was formed in a club on Bridge Street in 1893. The founder members included Caley Clach and Thistle all of which had their origins in the 1880s. In contrast Dundee FC was formed in 1893 while Dundee United are very much New Kids on the Block, having been formed in 1909. Until the 1930s there was also a fourth Inverness club, Citadel, in the Highland League.
For allegedly not having a football tradition, Inverness seems to have done rather well, finishing for the last two seasons ahead of both clubs from the City of Discovery.
I suppose Inverness can also lay claim to being a "football city" on the strength of long term support here for Celtic (founded 1888) and Rangers (founded 1690).