What limits the number of missile engagements a ship can support?

Is spreading missile defence across several ships partly a response to the number of targets?

Yes. Computers, targeting data, radar numbers and interference limit engagement capacity, while concentrating more systems in a very large hull creates difficult and expensive design work.

Carrying missiles does not by itself provide the equipment needed to control their engagements. Computer capacity, targeting data and the number of radars impose limits. Radars can also interfere with one another, making their arrangement part of the naval-architecture problem. One objection to converting a battleship or Alaska-class hull is that radar interference would prevent a missile fit large enough to justify the ship's size. The counterargument is that careful naval architecture could permit the fit, although the work could be financially prohibitive.