What two three-digit numbers, when multiplied, equals a six-digit number...with the product containing all the same digit?

Well the lowest such 6 digit number would be 111111
A bit of fooling around determines that 111111 = 3 x 7 x 11 x 13 x 37

if we divide this into 3 x 7 x 11 = 231
and 13 x 37 = 481

Then 231 x 481 = 111111

You could then multiply either 231 or 481 by 2 and get two numbers that multply to make 222222
Or multiply 231 by 3 and get numbers that make 333333
or mulitply 231 x 4 and get numbers that make 444444

or multiply 231 x 3 and 481 x 2 and get 666666

and there's probably some other combinations that work, but I've got to go now.

Pick a six-digit number, say 444444 and find the prime decomposition:
444444 = 4*11111=4*3*37037=...=4*3*7*11*13*37
Now try to think about how to combine these into two three digit numbers. Obviously 37 can't be combined with 13 and 11 because the product would be greater than 1000.
You'll find that
4*11*13 = 572 and
3*7*37 = 777
and 777*572 = 444444
Try it with another number!

I would have to try all one million possibilities.

Unless you mean something simple, like 143 x 777 = 111111. The product is 111111, and all the digits are the same -- they are all 1's.