Minimally more data.

A normal alpha character is 8 bits (1 byte) long.

Standard emojis are encoded in UNICODE. There are a couple of versions of UNICODE, but the most used one us UNICODE-4, meaning that a UNICODE representation of a character can be 32 bits (4 bytes/characters) long.

But a byte is a byte (and an emoji is emoji) whether it is on a cell phone or an internet line from your ISP. An emoji is going to be 4 bytes long no matter what. Cell phone doesn’t matter.

The ISP person was (a) full of shit, or (b) lying to you, or© utterly ignorant.

An emoji is considered a text message and therefore charged as such.
There are video emoji“s which I wonder cost more ?
I stopped using emoji“s within my texting as it caused problems.( text multiplied and I thought that I had a virus but customer service tech told me it was too many emoji’s within a text that caused the problem) . I figure it could had been slow cell service..but since have upgraded the system and all OK.