I'd happily pay a dog licence fee and hand over a DNA sample of my dog. I'm as fed up of the next person of stepping in dog shit or not being able to take my dog to places where they're now banned because of the irresponsible behaviour of other owners. I don't think a DNA register will change much though.
The dog poo dna testing scheme has been piloted in several areas in the uk, but hasn't caught on nationally for several reasons.
Firstly, DNA is tricky to extract from faeces. Doable, but more expensive and easily contaminated. Has a small forensic window of time to be collected, too. You also have to be able to match the DNA. The kinds of dog owners who let their dogs shit everywhere aren't going to offer their dog's DNA to a database voluntarily. Currently there's no way to mandate that or enforce it if it's flouted. The odds of being caught would always be minuscule.
You'd have more misses than hits meaning a significant cost for the local authority. You'd also need a significant increase in the number of dog wardens to make an impact. That's expensive and not recoverable through the existing legislation that allows a £100 fixed penalty notice up to £1000 fine if not paid within time.
You could make a licence and dna sample mandatory and increase licence fees and/or fines substantially, but that would require new legislation and fines are always set within established parameters. You are not ever going to see a £50,000 fine for dog fouling. So the revenue will always be limited.
A licence fee is unlikely to be a vote winner, especially given the number of elderly on fixed incomes who fall into that bracket and many charities and lobby groups will argue it's unfairly penalisong the poor and/or vulnerable for having the temerity to own a dog, especially when we know that caring for an animal and taking regular exercise is good for mental and physical health.
In today's transactional society, people will also expect something back for their licence fee, such as more dog bins, street cleaners, dog walking areas and more wardens to catch that minority who persistently ignore it. All of which will eat into the money gathered substantially.
If you want to seize a dog from its owner, you'd again need to change the law. Antisocial defecation doesn't fall under any existing powers to seize a dog. What do you do with the confiscated dogs? Dog owners who let their dogs shit everywhere probably don't train them or socialise them properly either, so they'd not be easy to rehome. So are we talking euthanasia?
Far better to win hearts and minds. The last time we ran a pr campaign on dog fouling it reduced fouling by nearly half, and up to 90% in some areas. That's infinitely more affordable.