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It makes sense that he keeps his original name since he plans to leave this world. His whole goal is to do to dxd in which case why would he completely abandon his original name when as soon as he gets to the next world he will use Ian again. It makes sense though to use this worlds name when in front of non friends as it would be weird to randomly use his original name.
Story could probably use more time developing inter character relationships. Not every chapter has to have some big fight event for example it could just be a date with Laura or growing the friendship with the other avengers. Imo it would feel bad breaking him and Laura up as with their mutant powers who else could they be with they will both have such long lifespans and they haven't even had a decently long conversation. Food for thought I guess.