Tile floors grout is coming up where dogs run

Hello, I have a two room each being roughly 9x9 with 12" tiles throughout. I remember when laying it down I think we had the mix to dry or something but I dont really remember it was 7 years ago. Anyway, we have 2 German Shepherd dogs and they always run through and around the two rooms (Thy are only separated by a tiny wall) and where they go they have literally tore a lot of the grout down to the mortar. Should this or could this happen? Next I want to regrout but I need to know what tool out there is the easiest for grout removal because I might end up doing the whole floor. The lines are 1/4" and how deep do I have to take the grout down?

Thanks!