The Navy and Starfleet have many organizational similarities, and the rank structure for commissioned officers is the same.
The rank titles are the same but that doesn't tell us anything about the timeline of a career in that service. The US Navy had the same rank structure in the 1880s as today, but back then it was normal for a lieutenant to have over 20 years of commissioned service.
Judging by Kirk's academy classmates in "Court Martial," who are lieutenants, it does seem that Kirk is well ahead of the curve for making rank.
In TNG "Datalore," Data implies it normally takes 12-15 years after the Academy to make lieutenant commander.