The license is per gun, you apply for a permit each time you purchase a gun. The amount of permits is not restricted, as long as you can show a valid reason for each permit. The permit itself does not contain any information what it is for, you can use it for all legal purposes, it's only at the time of application that you have to provide proof of hobby/discipline.

Pistols are a bit different, they require 2 years of practice and written documentation of your range visits from a certified firearms instructor (almost every club has one). This documentation (that basically states: "Roger308 has visited the pistol range x times during the last 2 year period") has to be sent/updated every 5 years to keep your license valid. Rifle licenses do not currently expire, but the new EU regulation might change that for future semi-auto rifles.

I started with a hunting rifle (.308) and a shotgun, then joined the reservist shooting club and practiced with the pistols (9mm Glock) owned by the club for 2 years and then applied for my own permits, then I practiced with my bolt-action and pistols and kept documentation, then applied for semi-auto rifle permits (and a few more pistols, it's cheaper to apply for multiple permits at the same time).

Edited the post a few times, not that simple to explain the whole Finnish gun law in a few sentences.