The modern induction type electromechanical watt-hour meter was invented for the Westinghouse corporation in 1894. Prior to that there were several different designs for metering electricity running back to Samuel Gardiner who invented a meter that measured how long electricity was applied to the load (it didn't measure how much power was used just when power was used) and to Thomas Edison who in 1881 developed a meter for his DC power system.

Read all about it here

In the US metering began very early, coin operated meters were not uncommon and were popular in older urban buildings for sub-metering purposes. The utility meter reader would collect the coins on his rounds.

Utilities were very 'consolidated' the company I currently work for grew by developing trolley lines to new suburbs, running the gas and electric to the area, financing home building then selling appliances, electric, gas and transportation to those new homeowners.

For what its worth unmetered, flat fee service still exists for niche markets (area lighting and various body politic services). On the other side of the spectrum you have interval billing which fluctuates minute to minute and measures not just total KWH but capacity factors as well net billing which allows selling back into the grid.

What has definitely changed is the format of the bill. I've seen many 60+ year old electric bills (apparently at one time it was fashionable for people to save the first electric bill after they bought their first home). They would just display your current and previous reading and the total amount owed. No rate breakdowns, no explanations or long complex fee structures just a simple PAY X by Z. They are about the size of a postcard.