What Is This Electrical Connection? (2005 America)

Looks like a connector for a battery charger/tender to me. Unusual place to locate one.

 

There is a standard loom for these bikes whichever part of the world they are destined for.

I therefore can’t say for certain without seeing the wire colours. On the RH handlebar there is a similar spare connector tucked into the switch housing for those markets where manual light switches are fitted.

On the LH handlebar I wonder whether your clutch switch been fitted/removed?

Try this
https://www.triumphrat.net/attachments/bonneville-wiring-diagram-jpg.4994/

 

Clutch switch I'd say

 

Poss heated grips. But need to see colour codes. That is a 2 pin molex. Only time triumph used them are heated grips and some indecators.