What are signature weapons?

Signature weapons are version of normally available weapons with pre-installed mods and custom paint jobs. Each of the base weapons is or will be available separately in the store, not necessarily before his "signature" counterpart (Specifically, the Vektor is available late game, I unlocked most signature counterparts before their "normal" versions).

You can check what criteria is needed for signature weapons by checking your hand book, they appear in the game progress summary page as stars (check the legend at the bottom), or this handy list (copied from here):

Signature Weapon - Condition Req'd to Unlock
Shadow - Liberate 17 Outposts.
Shredder - Collect 10 Memory Cards.
Bull - Collect 10 Relics.
Bushman - Activate all 18 Radio Towers.
Ripper - Complete six Trials of the Rakyat.
AMR - Collect 20 Relics.
Japanese Tanto - Find and collect six Letters of the Lost.
Cannon - Purchase from UPlay

The weapons will always have the same mods/color scheme, regardless of play style. Three of the signature weapons - Shadow, Shredder & Bushman - have suppressors; three weapons have red paint jobs - Shredder, Ripper & AMR. I'd double check your facts, because of those only the Shredder is both red & suppressed (AMR is anti-suppressed, TBH...).