I didn't really have my own pet til adulthood, but I did think of the family's or my dad's dog as my own. I consider my dad's my first, because he's the first I remember and the first that caused me to deal with loss when he died.
"Goopey Vincent"
Did I mention he was my dad's dog?
Also, as to the safeness of the street name, I don't really see the problem unless you have compounding information "out there." Street names are fairly ubiquitous, at least here in the states, especially if you include the various types of road: street, road, drive, lane, et al. I seem to remember being in a city that had the same name attributed to a street, lane, and road, all at different locations. Glad I wasn't a delivery boy there.
But I can see the worry. Better to stay safe.
Who the fuck names their pet "Goopey?"