I am going to come at this from a perspective that is opposite of yours. In my situation, I am married to the father, while SS resides full time with BM and her GF. 

BM moved SS out of state to be with her GF. GF immediately started taking on the role of other parent and in many ways overstepped in that role. It started off with little things and eventually grew to big things. She was listed on all of SS' school and medical paperwork, she was taking SS to all medical appointments, signing him up for daycare and sports, etc. She took on the role as parent, even though it wasn't in her right to do so - BM was not married to her and GF was not SS' stepmother. This started interfering with SS and DH's relationship because SS saw them as a nuclear unit because of how normal it all appeared in the day to day and BM worked really hard to make their family (BM, GF, and SS) appear nuclear and whole. Because of the distance, it was really easy for them to act like DH doesnt exist and that they are a complete family. SS eventually started calling GF "Mom" and to this day, keeps saying "I have two Moms."

While you are a very important part of this little boy's life, you are not his parent. You are not his "other father." I know how extremely gut wrenching that is because I have always wanted to be SS' "other Mom." Sometimes when you want something to be true, I think you nudge it to make it happen. Do you think about why you take on a lot of the day-to-day? Do you think part of that is related to your desire to be that "other parent" and to have a that family attachment with this woman and her child? 

It's obvious that this little boy loves you, but can being CCB be enough for you? I am my SS' stepmom, he calls me as CastleJJ. I have worked hard to accept that I'm not really a parent, nor his parent. I do assist with some of his day-to-day needs when he visits us, but I am more there to back up DH and be supportive. If you and your GF want more children and being a parent is important to you, focus on that and try to look at the current dynamics for what they are; that your SS has a mother and a father and two other awesome women in his life, all of which love and support him. You will get your chance to be a parent to your own children, if that is what you want. 

I am sure that you do love your SS and that he loves you. You seem to really be invested in his life which is wonderful!

Keep in mind that the bio dad does get to be the one called dad even if he is sometimes a deadbeat dad. There will come a point in SS's life where you will be a villain in the whole thing, so will his other SM. Teenagers and Preteens are THE WORST. 

I would suggest coming up with an alternate nickname for you. Do you and your GF have plans to marry and make you legally permanent in their lives? You might want to just stick to your name until then. It would help you guard your own heart. 

It might be wise to start thinking of yourself as more like an aunt to him, rather than another parent. Turn over the day-to-day stuff to your GF and the boy's father, and you get to just be the fun one. You can have a really positive relationship with him without being another parent, and it might help you define your own role better and detach a bit.

I'm going to "attack" this from a different angle, specifically from the jealousy that your partner may still have feelings for her ex.

Digging deep, what is the reason for that? Is it because she and her ex have a too-friendly relationship? Is it because you lack self-esteem? Is ir because you have a bias against bisexual women?

Jealousy happens. When your childless and want to be a parent, those feelings come on strong. You tackle those feelings, though, by examining the cause and unpacking that. If your partner is being too friendly with the ex (e.g. they flirt, they hang out, she gets depressed when he leaves), then you need to talk to your partner about how that makes you feel. If you have self worth issues, you need to talk to a therapist to learn how to build that up in yourself. If you have a bias against bisexual women, you need to dig deeper into that and ask yourself why. 

Once you figure out why you're jealous, you can start taking action steps to dismantle it. And you'll need to take those steps or you'll end up resenting your GF.

I'm new here and am going to chime in with what will probably be an unpopular opinion. Before I do, I want to be clear I'm not advocating for any stepparent to usurp a bio parent's role or be called mom or dad. There is another aspect to this, though, in that this isn't a case of protecting or doing what's right for the kid, it's about placating an adult who should get over himself and be happy his child is well cared for in his absence.

We seem to have this idea that the kids don't want another parent in their lives. That's often true and we shouldn't overstep. However, there are also cases where that's learned behavior. Kids usually don't judge, at least at first. Hatred, prejudice, defensiveness is all often learned. In this case, a child, apparently on his own, identified a loving caregiver in his life as a parent because that's the role that was filled. The child has no problem with this. If he has a problem with it later, I'll bet he learned it from his dad telling him it's wrong to love someone else as a father, not because the kid actually feels that way. It will be learned predjudice.

In this case, it is the adult that's sensitive. The adult that has an issue with not being the one and only daddy even though he's often absent. The adult that we must now placate because he's being sad and sensitive. That's ridiculous. Unless BM chased him away with court rulings, etc (which is possible) then daddy dearest is the one who is choosing to leave space in his son's life for another father figure. The person kindly filling that role shouldn't be punished for that and told to stay in their lane.

So yes. Work on your jealousy toward bio dad, because his relationship with this kid is important. Work on your jealously over your GF and his relationship. Those are your issues to overcome. But congratulations on being a strong parental figure in this child's life. Congratulations on having a stepchild who is open hearted enough not to distinguish between blood and plain love. You're doing a great job.

This happens when you get divorce.  There is a new SO. Who is going to have influence on his kid.  That his problem, his problem to work out between him and his DS.  What will happen if he gets a SO ??  How is the new SO going to fit in?