Zeuhl is right. This is a trademark issue. If the other person has registered their name as a trademark (unlikely, but still possible) then you will have to back down. Strictly speaking, trademarks can be duplicated if the businesses are distinct from each other, but if you both do some sort of dog care (even if not exactly the same) then there is a potential clash. Again, it also depends on how near you are to each other geographically - if the other end of the country, there should be no problem, but if within a reasonably accessible distance for someone wanting the service, there is again a clash. Many people running small businesses do not register their names as a trade mark due to the cost (not astonomical, but significant if the budget is small).
If the name is not registered, then it can still be claimed to belong to the other person if they have used it for a reasonable length of time (I forget the exact times, sorry). The rules can be found on this website: www.ipo.gov.uk/tm.htm. Zeuhl's point about goodwill is the issue here.