The Anti-Monitor takes a form completely different from his comic and live action counterparts; being a jet black being, rather than be a living armor or some kind of shriveled pale being.
It makes sense Terry calls out the Tommorowverse Batman on refusing help and choosing to be alone. His Batman had a Bat Family and only ever let them go after extreme circumstances; Nightwing couldn't handle Bruce hurting criminals in front of their children, Robin was driven insane and experimented on by Joker, Barbara suffered a miscarriage while crime fighting and broke up with Bruce. At least in his old age, Bruce had a dog and accepted Terry's company as the next Batman.
It's not explained how the Monitor, nor the Flash by extension for creating the new timeline, allowed the Anti-Monitor access to the multiverse.
The fact Terry McGinnis/Batman Beyond is assisting with guarding the tower, he was likely plucked from Earth-12's future (2038). The trailer for part 3 shows Batman and the Joker fighting in between the 1990s and 2000s; which would place it concurrent to the Crisis.
This is the second adaptation of "Crisis on Infinite Earths" as it was previously adapted as a crossover between the CW's DC shows as Crisis on Infinite Earths: Part One (2019), Crisis on Infinite Earths: Part Two (2019), Crisis on Infinite Earths: Part Three (2019), Crisis on Infinite Earths: Part Four (2020) and Crisis on Infinite Earths: Part Five (2020).