Two different silkscreen colours on the same PCB
There is no technical reason why it couldn't be done, though it would not fit with some production processes where boards from different customers are panelized. Often the silk screen colors offered are very limited (white or black only) so they may not be set up for what you want.
Just ask the PCB maker, and expect an additional charge and possibly longer lead time if indeed they are willing to accommodate you.
You will have to supply separate Gerbers for each of the silk screen layers, of course, and a readme file that explains what you want. Also describe the order in which the layers should be printed, if that matters. Keep in mind that the registration won't be perfect so overlapping or touching colors have to be dealt with as you would with any another printing process (and the silk screen registration is typically not great compared to regular printing). In graphic arts parlance, this is called trapping (image below from Adobe link).
In the above, a certain amount of overprinting A is deliberately caused (equal to at least the possible misalignment between silk layers) in order to avoid the background showing through around the knockout area B.
You are essentially creating color separations, so these problems exist if you have more than one color touching.
It's easy enough to silk-screen print a board yourself, but the inks used that resist common solvents can be troublesome and/or expensive in small quantities.
For fans of Ukiyo-e floating world prints, here is an example (after Hokusai's print).
And a more functional one done using OSH Park's standard colors (by a denizen of this site).
That is out of the ordinary process for board houses. There is no standard way to specify that, and many board houses may not even want to bother, or charge you a lot extra. The only way to find out is to talk to some board houses to see if any will do what you want and for what price, minimum quantities, etc.
Another possibility is to have the board house use normal white silkscreen as usual. If you want a different color somewhere, specify a full white region to the board house. Then paint that yourself if you have the process, affix a pre-made sticker, etc.
Sounds expensive, but certainly doable if you can find someone willing to do it. It would be cheaper to use one color for silkscreen, and then get creative in the soldermask layer. You could also use different patterns and crosshatches in the screen layer.