I realized my original post had some wacky font size formatting so I fixed that. Sorry.
Now, you can apply a second coat, but it won't make any difference in intensity of color. If you are applying the first coat to bare wood, it will soak in a little, but the second coat will not melt into the first like lacquer or shellac, because while there is shellac in the ink, it's waterborne, not contained in a highly volatile solvent like acetone or alcohol, which achieves its bond to the previously-applied layer by melting it somewhat. I'm not really sure how it will behave from the standpoint of bonding to the existing color coat, but experimentation on scrap is always worthwhile. My own experience was that I didn't need any more ink after the first coat. I used Speedball ink on my project as well.
Side note - India ink that is otherwise sanded back to bare lumber remains in the grain and gives great contrast under a colored dye coat.
Example from my very first finishing project ever. This project was coated with India ink, then I had a change of heart and sanded it back. I then dyed with full-strength water-based General Finishes dye stain, sealed, and then filled the grain with ebony Timbermate waterbased grain filler. Sealed with Deft rattle-can lacquer, then applied burst edge with Behlen black toner from a rattle-can, then lots more rattle-can lacquer. I think it's a pretty nifty look.