Laminator_X
Once & Future KMC Hawkeye
Get ready folks.
In DC continuity, all the Infinite Earths were splintered of from one main Creation by Krona's blasphemous research into its origin. The original world itself was mainly cleaved in twain, a Positive Matter universe centered on Oa, and an anti-Matter Universe centered on Qward. The Monitor and Anti-Monitor were born of this accident as well.
The Anti-Monitor's ability to consume these worlds (as I understand it) arose from their common origin in/before Krona's trespass. By the time he came for the "original" positive universe (the one with the JLA, not the JSA) he had absorbed so many of the splinters that the Monitor was no longer a match for him.
In the Marvel continuity, before time as we understand it, there was an All-in-All mega god who was the only thing/being that there was. It eventually grew bored or something and decided to cease to be. The fracturing of this being's essence gave rise to the multiverse, with all its different cosms, cycles of expansion and collapse, magic realms, and what have you. The Infinity Gems were focal points for the remnants of this being's essence. It is over these worlds that the Living Tribunal holds jurisdiction, whose structure is kept fast by Atleza, and where the Phoenix Force catalyzes cycles of destruction and creation.
These are not part of the same continuum at all.
I do not think that the Anti-Monitor could absorb the MU, because they do not arise from the same prime origin. This would also explain why the IG didn't do anything when Darkseid found it in JLA/Avengers.
The Anti-Monitor may have sufficient raw power at his disposal to lay waste to the a universe by force, but that's not how he was doing it in Crisis. If I had o guess, the LT could simply bar his entry to the Multiverse that is his to protect, or, as he once thretenned when Zom's lock was cut, to destroy a Universe that the AM invaded in order to keep him from spreading his contmination from spreading to others.