If each of the five factories that share the electricity network is built, additional factories will not receive electricity. When the sixth factory is built, the message "five connections have been reached," appears. Is this intended? Or was the problem caused by the decrease in entities?

There is also an additional problem, and if the above situation occurs, the new factory will not continue to receive power until all five of the other factories are removed. (If even one of the five factories remains, it cannot be supplied with power.) Even if a new factory is added after demolition of all factories, the message "five connections" occurs from the second factory, and power supply is not available.

the building connects to the power grid by linking power-poles across surfaces.
this creates an invisible cable connection.
a power-pole can only have 5 wired connections.
with SHIFT+click on the power-pole, you remove all wired connections (visible and invisible)

This is intended behavior and a modding api restriction. Only 5 factories are allowed per pole.
In vanilla, you can only connect 5 power poles together, and the same rule applies here.