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's not explained how the Monitor, nor the Flash by extension for creating the new timeline, allowed the Anti-Monitor access to the multiverse.
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.
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.
The reason the Tommorowverse Batman chooses to be alone may be due to subconscious echo of his DCAMU counterpart in his mind. After seeing all of his Bat Family died fighting Darksied's forces, except for Damian Wayne being revived by his girlfriend Raven, it would make sense that he would prepare a mental block before the reboot; living through that pain once was more than enough.