Scottish striker Miller led out the Cumbrians for the first time in Sunday's victory over Coventry when Abbott's side secured a third straight win at Brunton Park.
"I've never been a captain before so leading the team out was different for me," Miller told the club's official website.
"Obviously I'm delighted about it and feel very proud. The lads work for each other and we're all mates in there. There's a good spirit so hopefully we can keep that going.
"I had a discussion about it during the week with the gaffer and Danny (Livesey) and (club captain) Paul Thirlwell and I'll be the team captain now.
"Greg didn't really give me any specific instructions. He just told me to go out there, play the way I usually do and be a leader.
"There's more than one leader out there, even though I'm the one who has the armband, so we're all working for each other anyway."
Miller, sidelined for four months with ankle and groin injuries before returning to action last month, wants Carlisle to start picking up points away from home following three straight defeats on the road.
"We know we've been inconsistent," he added. "We need to try and get a foothold in games then defend well, like we did (against Coventry). Hopefully we've turned the corner and that can be the result which kicks us on."