Let me take a stab at answering this from the opposite side of the situation: I am a full time custodial parent (mother) who found a new partner when the child was 18 months old, said partner is now my husband and a full time care giver ("daddy" in every way except biological) to the first child and two more of the child's (half) brothers (technically). Ok, so now that you know my "blended" family status, you can see how I am qualified to provide some input here. I won't however, try to proclaim that our solution or my advice is broadly applicable to yours or anyone else's situation.
The answer to your first question "is it normal for the biological father to be excluded," is, generally, no. In some cases, where abuse or some other risk of harm to the child is a threat, the answer would be, understandably, yes. But, that's not the case here, otherwise, the mother would be seeking ways to exclude you in more ways than just attempting to strip your title.
The answer to your second question, "how do I persuade the mother to let the child call me DaddyName" is you can't. Sorry, but you most likely will have to agree to disagree. You will find, if you haven't already, that there will be at least a few issues that you two won't see eye to eye on (which is why you aren't together, right?)
One of the things that works both for you and against you here is that neither of you can control what goes on in the others' home. I mean, with the exception of something dangerous or illegal where child protective services would be involved, or the police. We're not talking about that now. Of course, you guys should work together to have a fairly consistent set of rules at both homes, but, let's be real: the rules, routines, norms, and traditions in each home will vary slightly. This is normal, unavoidable, and not something your child won't learn to navigate. Now, this means that at her house, you can't make her allow the kid to call you DaddyName. Maybe you'll be just Name. However, on the flip side, this also works in your favor because she can't force the child, when s/he is out of her control to call you anything other than DaddyName, or, whatever you and the kid work out. Btw-have you considered "papa" or some other alternative? Because it IS confusing to a four year old to have two daddys, which is why DaddyName works well, but might not offer enough distinction to a young child, or to the child's siblings if he has any.
I agree with some of the other answers here that suggest that she's doing it to try to provide a "normal" family environment for your child, especially if she has children with her new husband. It's nice to see that you aren't jealous of the relationship with the man who spends quite a bit if time raising your child. I am lucky enough to also say the same is true for my son's dad. In my house, my boy initially called my husband Name. Eventually, he called him "DaddyName" and now he calls him just Daddy. He had always called his biological father Dad at both houses, but began calling him DaddyName at ours when 1) his first brother was born, and 2) when he started calling his step dad DaddyName. For a while, both dads were DaddyName at the both houses, but, it has now completely reversed so that at our house, his biodad is now Name, and step dad is Daddy. He also calls step dad Daddy at his bio dad's house. It irritated him somewhat, at first, that he is Name at our house but he has enough sense to realize that at this point, my husband IS his dad, and like you he is (mostly) grateful that his boy is loved and cared for. Also, he can see how it's confusing to my boy's brothers (who are very young) that there are two "daddys" (is he my daddy too? Does everyone have two dads? How come we don't get to have two dads...and so on.)
We did, btw, years ago have the conversation about the names. He was agitated not by me telling the child to call him or my husband one thing or another, but that I wasnt instructing him at all . I told him I would never "instruct" our child to call either of them anything. If the boy wanted to call one "Monkey" and the other "Goose" it didn't matter to me at all because it had zero impact on the quality of the relationships. I figured the kid could figure it out on his own-letting the relationships guide him. I advised my son's dad to not push too hard, because children sort of naturally do what you don't want them to, and warned it would backfire. I told him, politely, what I said earlier about him not having any say in what goes on in my home, and vice versa. When put that way, he backed off, to sulk at first but he came around.
Ultimately, his distaste for the boy calling his step dad "Daddy" stemmed from his insecurity as a father. Once he figured out that no matter what, he IS the boy's dad, and the boy loves him as a dad unconditionally, he was satisfied to let the name issue drop. I'm not saying you are insecure, but if you are, don't be. As long as you are there for your child, that you love your child, care for him, he will know who you are and will call you by your name. He'll figure out (when he's a little older than 4, be patient) how to navigate both homes' rules (with frightening cunning at some point-Beware!). He'll call you what he'll call you even if it displeases his mother, at your house at least, but hey, that's all that you care about right?