I saw a report on TV that food coloring can be animal-based, especially the red ones.
Is it the same for common printer inks (i.e. printer brands inks)?
If so, is there a vegan alternative?
Ask the manufacturer.
Printer inks do use some organic compounds, and those could very well involve animal products at some point in the processing, but each manufacturer has its own recipe, and often tweak the ink to work better with specific types of printers. Some might even pursue some kind of vegan certification, in which case they would probably label it visibly.