'65 Dart - Is this a fusible link and what is the brown wire for?

Not likely; But it could be a "poorman's" F-link.

A fusible link is just a short length of multi-strand, usually two sizes smaller than the wire it is connected to, so that, if a high draw occurs, this wire is sacrificial, and it melts, before your car catches fire.
It is more useful than a fuse because the fuse would blow instantly. and
More useful than a automatic resetting thingamabob, cuz the power may not be interrupted, as, F-links can be heated many times to below the melting point, and if you become aware of the problem upstream, you can fix it before you get stranded.
What that means to you is this;
Any wire can serve the purpose of the F-link, when it is appropriately sized for the circuit. And out on the road, guys do it. I do it. But when I get back home, I splice in a properly tagged link.

BTW, on 67>72 cars, the ones I am familiar with, the two wires on the starter relay are always Yellow and Brown, if that helps and neither is fused. One of them, IIRC the Brown, is coming from the start circuit, which comes from the ammeter, which comes thru the bulkhead connector, where is is fed from the battery thru, you guessed it, a fusible link.
That's how I remember it. I could be wrong.
The yellow IIRC, goes to the NSS on the transmission and just grounds the relay in Park or Neutral.
In 50 years, I have never seen a problem in this circuit, and in fact, the relay itself on my 68 Barracuda is likely original.
But, I get it, chit happens lol. and then you get a splice, lol.