The current standard defining the size of credit cards is ISO/IEC 7810, last revised in 2003. If you visit that site you can look back through prior revisions; the earliest I can find relating to credit cards/magnetic stripe cards is from 1976. (You can also preview the current revision of the standard).

I don't have access to the text of the previous standards unfortunately, but I would suggest around that timeframe for the "standard" per your question. Credit cards certainly existed before then - Diners Club, American Express and a few others started as early as the 1950s - but I can't find any indication of what sizes those cards were.

EDIT: American Express started out with paper cards but was the first to switch to plastic in 1959. (Source) That would probably be a good starting place for a deeper study.

Is there a relationship between credit card size, business card size and/or poker card size (which are all about 2in x 3.5in)?

Diners Club was amongst the first (or the first depending on your definition) credit cards being in operation from the late 1940's. There are some images here dating back to the 1950's.

http://www.dinersclub.com/press-room/card-history.html

Unfortunately no sizes are given but the proportions appear to be the same as today and it seems reasonable to think the size was then more or less what it is today .

Here's a site showing the evolution of Diners Cards over the past 50 years: http://www.dinersclub.com/press-room/card-history.html

Notice that it isn't until the late 1970s or even the 1980s that the current card design becomes common -- early cards did not have the current 16-digit embossed numbers, nor they have the magnetic stripe on the back. These features only became common in the 1980s.

The best recent examination of the economic and development of the modern consumer-credit industry is Evans and Schmalensee Paying with Plastic: The Digital Revolution in Buying and Borrowing (2006). This is more of a business and economics book than a history book, but it still gives a good analysis of how the credit card industry grew into the billion-dollar behemoth it is today.