What you are refering to is a process known as "pad printing". Pad printing uses a flexible silicone rubber pad presses down on an ink covered printing printing plate and back down on the item to be printed.
It's a slightly similar concept to say if you stamped a rubber stamp onto an ink pad, then onto a piece of paper - except instead of the design being part of the stamp it comes from the printing plate.
A slightly complicated explaination of this can be seen here: http://en.wikipedia.org/wiki/Pad_printing
Generally this is limited to spot colours as each colour is printed seperately. There are machines available for 1, 2, 3 or 4 colours, with 1 and 2 being most common. While pad printing can do very tiny details on some difficult 3D products (think tiny type on a golf ball), it's not suited for full colour logos nor logos with a lot of shading in them.
A couple other points - generally it's only suited for smaller items. In most cases products need to be solid rather than soft or flexible. The exception is squeezable stress toys which have special techniques and skill employed by the operator to print them successfully.
Also of interest is that you mention "most likely printed during manufacture" - In reality most products of the type you mention are probably printed at the point of manufacture using either the exact process I mention or a similar process.
Re volumes, this is an economical printing process once you get up to a reasonable volume, but isn't really suited for very small quantities. A small volume for this process would be anything under 100 units. The reason for this is once the operator gets going they can process products quite quickly, however the process involves a LOT of setup time - to print 100 products may take 1.5 hours, but to print only 1 would still take an hour or more. Note also that each different product would usually be considered a different job - with this printing process it's not really possible to do 5 each of 20 different products and call it a 100 unit job!
If you are located in Australia the company I work for (Global Advertising Products) is one of the few promotional product companies with in-house pad printing facilities. You can contact us on 1800 060 305, or website http://www.adproducts.com.au
. If you are located in the USA or elsewhere I can't specifically recommend anyone, but if you do a search for "pad printing" on yahoo or google you may find someone suitable.
Best of Luck
Global Advertising Products