It all depends...
Most kit will either disconnect the internal battery when an external one is plugged in (not a problem), or try to charge the internal one from the external source. That's not a problem if both are 12 volt (the Icom is), because no charging current will flow. But if the "true" voltage of the internal battery is lower, then the internal may run the external down attempting to charge itself.
More seriously, if you have normal dry cells fitted and the "system" attempts to charge these, the results could be quite drastic (explosion is theoretically possible). Again, you're safe with the Icom because the dry cell pack doesn't have the extra connection for the charging line.
Usually, there will be a blocking diode in the connection from the external power supply, which reduces the voltage on offer from that source. In this case, the internal battery will be used first, until its voltage drops to that offered from the external.
As a general rule, NEVER run any equipment like that without the internal battery. If the operating voltage is lower than the external supply (as is often the case with equipment that charges the internal battery from the external supply), then the peak voltage - for example on engine start or when using an external power supply - can fry the equipment.
I once saw all the electrics and electronics on a twin damaged by someone connecting a booster external power supply with the battery master turned off. Unfortunately, the external PSU was offering about 70 volts off-load to a 28 volt system.
So, with the Icom it doesn't matter at all. I don't know the GPS 55 so can't comment on that.