OP, I have no idea on "staining", however, I have used a product called Grout Shield, a specialized grout sealing product that can color (or color match) grout while it seals the grout. I have used this in the entire house sealing the floor and bathroom grout, and it works very well. This is primarily a DIY product. I did a Gardenia, about 1600 square feet tile and to bathrooms, one with a walk in shower, very happy with the results.
NOTE: this is a paint like material and is designed to match existing grout, but can be used to change color. I have no idea if the "stain" provides a sealing function, I would check on this before using.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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