The quick answer is … yes and no.
Consider this. How many printer models do you think there are out there right now? 1000? 2000? Who knows. But one thing I do know is nobody is going to spend the time testing the wrong inks in the wrong cartridges all day long to see if it works.
All printer manufacturers do things a little different. Without getting technical, there are basically three different technologies for inkjet printers. Epson uses Piezoelectric, Canon uses Bubblejet and HP and Lexmark use thermal inkjet technology.
An ink designed for Canon, for instance, has a much lower convection rate than an ink made for HP or Lexmark. The reason is HP’s cartridge print heads fire at thousands of degrees. The ink must be made to withstand that kind of heat or your resulting output will be unpredictable.
Also there are basically two kinds of ink. Dye based and Pigmented. Pigmented ink particles are much larger than dye based particles so using a pigmented ink in a cartridge designed to use dye based ink will result in a clogged print head every time. Also, pigmented inks are waterfast on any surface. Notice I said waterfast and not waterproof. Pigmented inks can made waterproof if they are used with the proper media. The same goes with dye based inks.
Pay attention to this because it will apply to any ink on the market. Any dye based ink can be used in ANY inkjet printer. The color output may not be what you expected but it will work.
Not so with pigmented inks. Pigmented inks are used in most black inkjet cartridges today. Epson has a version of color pigmented inks they call DuraBrite inks. They are very expensive to make and there is quite a hefty premium on the bulk inks from any manufacturer that I have contacted.
DO NOT use pigmented inks in any cartridge that was designed to use dye based ink.
Generally speaking, HP and Lexmark inks are interchangeable because their process’ are similar. Watch for color variations though.
The newer Canon cartridges, BCI-3e and BCI-6 colors are so close that most people cannot tell them apart. Ironically the cartridges are physically identical except for the BCI-3e black which is slightly larger. The BCI-3e black takes pigmented ink while the BCI-6 takes dye based. Be careful here. Some of the new Canon printers take both BCI-3e and BCI-6 black cartridges.
To wrap it up I’ll say that you can use most inks in most printers with the exceptions that I already mentioned. Color variations, if they occur, might be compensated for in the printer driver settings. Be prepared to fiddle with it. Physical damage to the printer is unlikely in any case, unless the cartridge is leaking when you put it into your printer. But you wouldn’t do that. Would you?