Subpanel next to Main panel wired the same in 1990
You dont need to rewire anything more than likely as it is grandfathered in.
Heres the basics of why.
They want to have a separate system to create a Grounded path and a CURRENT return path (Neutral)
They also changed the naming of wiring
Ungrounded conductors are commonly called "Hots" or Hot legs
Grounded conductors are Neutrals and Grounds
Current Carrying conductors are HOTS AND NEUTRALS
You need to understand the terminology then everything will make sense to you. the NEC is going on a Bonding and Grounding kick meaning EVERYTHING is Grounded, Bonded and Grounded/Bonded again for redundancy.
It used to be that these systems were interchangeable but now the thinking is to have a completely isolated grounding system.
Your main panel is bonded together to give it 3 ways to ground/return. One is your Service Drop neutral/messenger wire, the second is your water pipe ground wire and the other is your ground wire that runs to your electrode/ground rod.
On a subpanel you do not ground them to anything external other than the 4th wire coming in(when in the same structure) except when in a detached structure you ground to 1 (usually a ground rod) AND run an isolated ground back to the main, 4th wire (giving you a redundant system)
to answer your second question, If you unscrew the bonding strap, one of the Bars should be detached from the case (Neutral) and the other should be connected at somepoint (ground bar) You need to make sure you have the right Branch Circuit wires running to it AND you have 4 wires coming in from the Main panel. Basically, when a panel is acting as a Main Breaker Panel, you dont have a "Ground Bar" you just have 2 neutral bars. When it is a sub panel you have a Neutral and a Ground bar.
DO NOT UNDER ANY CIRCUMSTANCE ALLOW THAT SUBPANEL CASE TO BE UNGROUNDED......
I hope this helps.
"Do it right the first time and avoid duplication of effort"
Your AHJ/Inspectors ALWAYS have the final say on ANY electrical code issue.
If in doubt, contact a licensed, experienced, reputable electrician to perform the work.