I am a step-parent, our daughters were six and four when I appeared, so I have experience on which to draw.
Your partner has no experience of children, and is feeling the pressure from all sides. Because he is not a biological parent, he senses a degree of resentment which is probably not there at this stage, but if he is not careful he will create it! He also will be aware, maybe not even conciously, that he is tne 'new kid on the block', and that you and your daughter represent a bonded couple - two against one. There is no rational basis for this - but this is how he will be thinking.
His response is to over-react, to see hostility and disrespect even where none exists, because he is hyper-sensitive. And your daughter will start to see him as an interloper who is upsetting your happy home, and the problems escalate. So how do you deail with it?
Point one - you must must MUST back him up in his discipline of your children, even if you think he is wrong. In front of them, you must be united in front of them, and discuss any issues between you out of earshot. Hand in hand with that goes the way in which you both discipline your children, and that takes teamwork. So, don't sit him down and lecture him on childcare - guide him into your ways, and affirm him when he gets it right, and guide him when he gets it wrong, and eventually, he will see that you are all one family, and it's not a matter of sides.
Just for the record, your own biological child will play one against the other - that's what children do! It's only with step-parents that other dimensions creep in - we have a third daughter, so I know this to be true.
While you are including him, include your daughter and get her onside. Explain that ytour partner lovves her, but he is new to being a Dad, and still learning, and she must be patient and help him to learn as well.
ctd ....