DSD needs reassurance and to be treated seriously, yes, but "treating seriously" has to mean dealing with this kind of behaviour, not indulging it OR ignoring it.
It is very typical behaviour by a teenage girl. Your DP needs to sit her down and give her a serious talk, and let her know that he loves her, and he loves you, and he - as you say - intends to make a go of it with you. Much better for her in the long run, I promise you, then ignoring it, making a joke of it, or letting her control when/where/how/if he expresses affection for you.
My teenage DSD has always enjoyed an extremely close relationship with DH. She had the same worries and reaction, and spent a lot of time, in our first year together, trying to mark her territory. DH did not indulge it. He was upfront with her about his feelings for her and for me. Thank god. In the long run, I thinkit has actually increased her confidence, because she now - over time - has seen that loving me has NOT taken away from her dad's love for her. If he had catered to her out of fear of upsetting her, I really think that uncertainty would have lingered. Think about tiny babies and separation anxiety - you don't solve it by never leaving your baby - you teach your baby that you leave and come back...that is the kind of real security your DP should strive for with his DD.
(BTW, I love when the 'it's normal teenage behaviour' line comes out. So is shoplifting, drugs and rink experimentation, lying, and all manner of disrespectful behaviour. None of those things, in moderation, make an abnormal or bad teenager. But only a confused or neglectful parent would allow those bahaviours to pass without comment and consequence.)
So that is my two cents...Your DP's DD is testing her limits and pushing the boundaries - DP needs ro sort it out, and your expectations/ideas are exactly right.