At some point you will have to take it to the vet, for vaccinations and a health check. If the vet scans it and finds it belongs to someone then you have problems.
Better to find out now before you get too attached to it (kids included).
Also you do not know its history or why it is straying, it may have some long term illness that the owners could not or did not want to pay for. Maybe it has temperament problems that will emerge later when it gets settled in.
Far better to hand it to the local warden now than let your kids get attached and finding out it has cancer or something and you cannot keep it. If it is a stray then the warden will allow you to have it back, but they will make sure that it is OK medically etc. first.
If it turns out to have a loving owner (who could be missing it desperately, it may have even been stolen and then abandoned!) then you can visit the local rescue centre and find a dog suitable for you and your family, there are loads of dogs out there in desperate need of a good loving home.
Good Luck, which ever way it goes you did the correct thing in picking it up!