As far as I know, all wireless devices work on an approved and licence free common frequency such as 433Mhz, but the communication protocol or in lay mans term the transmitted coded information differs, such that one manufacture will have his own protocol so that his devices don't clash or register with other manufacturer's, receiving devices, however if manufacturers were using generic or commonly available coding and decoding integrated circuits in their applications then this is where codes could be duplicated, some therefore make a provision to allow an end user to make small changes to protocol so that transmitted code is different to a similar device using a different code, they normally provide a bank of switches to make small changes to the code or so called channel setting, so likewise your bell may come with a channel set switch, and may have two or more channels to choose from, the idea is if your neighbour also owns the same brand of door bell, then his may set off yours and visa versa, so one of you will have to change the channel switch so that two door bells won't trigger each others door bell, same way your alarm panel that communicates with your external alarm bell may be getting triggered by an identical code from that door bell transmitter so if your bell has a channel setting switch you may select a different channel setting and it may no longer set your main alarm bell off.
I am assuming that it is not just the strength of RF Carrier from your door bell that is causing interference to your alarm bell thus triggering it off, that is possible but highly unlikely, otherwise your alarm bell would have been going off more frequently from other stronger radio waves transmitted from lots of other devices such as mobiles and wireless modems and so on.
Some wireless devices can be swamped by receiving continuously strong RF signals such that they cannot communicate effectively between a control panel and a remote device as the coded information gets corrupted by stronger signals and it is like you are trying to whisper to your partner whilst your mother is using a hoover!