I have either read or been told that the phone company can electronically split single line service to a residence and provide a second phone number resulting in two separate phones being provided through the same wire to the house. How do they do this?

Planning ahead.

The cable used to connect a home to the network typically contains four conductors. Only two conductors are required for standard phone service, so IF there a unused pairs leading the rest of the way to the switching office, installing a second line doesn't require any new cable runs on the phone company side of the Network Interface Device (NID). Often inside wiring to use the second line can be accomplished with a few swapped connections on the back of the wall plate as well. Dual line handsets would require a quick wall plate replacement.

In the past, most homes only had a single pair running into the house, and it is still common. For a while the phone companies were running larger cables, perhaps 4 pair, but as more people go to cell phones, the phone companies are cutting back again.

I don't think they can split a single analog signal into 2 lines, but easily could a digital line. I am not sure how many voice lines can be used with a DSL internet connection. Once multiple lines are split out at the NID or DSL modem, any phone an be plugged into any jack. Multiple line phones use the extra wires in the standard modular jacks.

They can, in a sense, do it with "distinctive ring". Distinctive ring - Wikipedia, the free encyclopedia

VoIP can also be used to have multiple lines, but your doing it over a DSL or data connection.

In may rural areas, there was a "party line" system in use many years ago. The would have a different ringing frequency. You could eavesdrop on the line and you would have to dial out using an operator.

We had an area down here where existing pairs to homes were not enough, The telco would install a, I believe their term was a "Multiplexer" which would have 2 voice lines in on one pair, and would have 2 pair voice out. This was a retrofit, they would remove the cover, and snap on a cover containing the Multiplexer. If term is wrong, I will research.