There’s always going to be a “my kid/you kid” dynamic in this situation because the situations are different. Unfortunately, because the father isn’t in the picture where his kids’ mom is, the OP will inherently have lesser authority over his kids than he will have over her son. Further, as others have said, the kid is six and you’ve been living together for 3 years with the biological father out of the picture, he inherently fills a much larger role.
Part of the problem is a respect and boundaries issue, but I think that’s sort of outside the scope here and those sorts of problems arise in some form or another in many relationships. You two need to discuss those, express your feelings, and work on them, and I don’t really have a whole lot of advice there. The problem I can give some thoughts on though is the discipline one.
Considering that he’s essentially filling the role of the father, you two need to sit down and discuss and come to complete agreement on how you’re going to discipline him. Obviously, as the mother, you get the final say, but it’s important that that sort of disagreement happens behind closed doors so he can get a consistent level of discipline. If spanking is something you’ll consider using, it’s not the sort of thing that can occur hours later, it needs to be reasonably close to the misbehavior or it is worthless. Either way, he needs to have a clear guideline of what sorts of punishments are appropriate for what sorts of behavior, so he can enact it as it’s appropriate, but he should also be absolutely straightforward and forthcoming at the end of the day and tell you what happened and how he handled it, so you know and can potentially adjust the guidelines.
This goes two ways though, as essentially a step-parent to his kids, being unwilling to do any sort of discipline for his kids probably isn’t the best idea. You need to talk to him about how you should handle it, but it’s going to be a lot more complicated because it needs to also be consistent with how their mom wants them disciplined. And, as you’re not filling that full-time mom role like he is the dad role for your son, chances are, your powers of discipline are, and ought to be, less.
All of this will, of course, result in some conflicts as your son will likely claim favoritism when he sees that they’re being treated differently, and you’ll have to find a way to explain it to him in a way he’ll understand, and I don’t really have a good suggestion there, I just think that failing to have consistent discipline will ultimately do more longterm harm than failing to have a satisfactory explanation for it.
That all said, I think it will be a lot easier for both of you in avoiding those sorts of situations if neither of you uses corporal punishment. I think it’ll be easier for all the kids to not get pulled in the the favoritsm problem if you don’t use that, since it’s a lot easier to put someone else’s kid in timeout or make them clean stuff up than it is to spank them.