That is an option and I've known people play this way, but it is more normal for us too play the same way you do and to discard 2 cards each.
We mostly play a two player game also, but our local Cribbage league, played in the pubs around the area is always a four player game, played in teams of two sat opposite each other.
The only other thing (which might be a local thing) is that we don't shuffle and cut between deals, the played cards go on the bottom of the pack and we deal straight off the top, however it is the dealers privilege if he wishes to shuffle and if he does then he must offer the cards to be cut, this was probably just done to speed up the game in pubs as it is one of the few games that is legally allowed to be played in pubs for money in this country.
The only other and obvious difference I can think of is the design of the board, which obviously doesn't affect the gameplay in any way.
Our boards are like the one shown here http://jesters.com/acatalog/info_ant973.html
Each player has a side, we score up the outside and then back down the inside, go round twice and finish in the centre hole.