Hi, and congratulations. Even an 'altered' dog...male or female...may 'mark' a territory that is being 'claimed' as their own. I had a female many years old 'mark' my brand new, freshly installed white carpet!! She didn't normally 'mark' anything, but for whatever reason...perhaps the installers had pets and transferred a scent...dunno...but she did it and was quite happy with herself for having done so.
Your new little Prince 'may' just be staking his 'claim' to the property, since he has just arrived to live with you and found another canine already lives there.
I have three Yorkies, all of which are definitely "mommy's babies" but they are also "daddy's buddies" so you may see some attitude changes over time.
I wish you the best of luck. Maybe you can post some pix of your little Prince...hint...hint...