Answer: The Rema (YD 92:8) writes that meaty and milky pots that touch each other do not contaminate each other. Thus, providing the lids are on, one can cook food in a milky pot next to that of a meaty pot. Thus, providing the knife and stone were clean when sharpening the knife, no taam, flavour, would be transferred from the sharpener to any future knives it is used for.
The Beis Yosef (YD 122:9) quotes the Mordechai (Avoda Zara 833) who writes that one who left their knives with a non-Jewish person to sharpen them must kasher them as they may have been used for non-kosher food. If one saw them being sharpened, however, and then took them home, they do not need to be kashered.
Based on this, R’ Shamai Gross (Shevet Hakehasi 4:192) writes that one may use the same knife sharpener for both meaty and milky knives.
The Shulchan Aruch (YD 120:4) writes that only items that come into direct contact with the food require tevila. Thus, trivets that support the pot don’t. Accordingly, a knife sharpener wouldn’t either.
In conclusion, a knife sharpener may be used simultaneously for meaty and milky knives and does not require tevila.