I typically use a muted, diffused or plain coloured background.  I can advise you on colours that will complement your pet's colouring.  I also understand that choosing a background might need to fit into your home decor.  I have painted portraits where there is a visible background, e.g. swimming in water or placed in a garden.   This may increase the cost of your portrait as it would take longer to produce.  This would be part of your requirement discussions.