You know what ? I'd love this to be true only because of the hilarity that would no doubt ensue as a result. I've said before that I never understood the reasoning behind turning Catwoman into Batman's one and only, solely based on the fact that both characters have had tragic pasts of some sorts as that's such a weak reasoning for a relationship as Gotham is full of characters with tragic pasts. What makes Catwoman anymore special than Poison Ivy or Harley ?
That's why I hate this narrative that DC and many rabid fans have created about how "special" Catwoman is, when there's really nothing unique or special about her or what she means to Bruce beside what the shippers insist on. To that end, I've been hoping to see other characters appear and challenge that whole idea, but DC seems smart (or stubborn) enough to realize how dangerous that would be and so they've seen it fit to basically prohibit Bruce from forming any sort of meaningful relationship with any female character not dressed in a catsuit.
Now that the background information is out of the way, what do I think of the possibility of a Bruce/Harley romance in the DCEU ? Well I don't think it's all that likely that this is anything but an easter egg, especially since I don't think Batman will be making any appearances in Birds of Prey or anything else. So whether the name on the dog tag refers to Bruce Wayne or Bruce Timm, I don't see it affecting the movie either ways. I'd imagine the idea of Harley hooking up with Bruce to be big enough to need addressing in the movie, especially since last we've seen of her, Harley was nowhere near Bruce. Also, the Suicide Squad movie made a point of mentioning that Harley helped kill Robin, so I imagine any redemption she could have will need a lot of explaining to do, especially if she's on a first name basis with Batman.
So in short, no I think it's highly unlikely that this is hinting at Harley and Bruce having anything together. HOWEVER, it IS an interesting idea to speculate about. Margot Robbie has proved herself both savvy and shrewd when it comes to matters relating to comics. She's really working hard to make her Harley gig into the biggest thing possible. You can see that with how she's chosen to go with a Birds of Prey movie and place herself in the spotlight, rather than share it in a Gotham City Sirens movie. I also distinctly remember her posting a picture of herself with a stack of Harley related comic books, with the most visible one being the valentine issue that Harley shared with Bruce.
So I can definitely see Robbie trying to steal Catwoman's spot as Batman's love interest away, I mean she's already done the same to Barbara. I can totally imagine the kiss that Harley shared with Batman in Suicide Squad being Robbie's idea. She really wants to make Harley into the biggest DC female character, and this could be her next step. I mean it worked for Catwoman for all these years and she basically has nothing going on for her otherwise, so I'm sure Robbie sees how beneficial having Batman in her pocket would be. But that's all speculation. Bottom line is, this sounds like something Robbie might push for, but I think it would be very difficult to pull off considering the circumstances, but I'd love to be proven wrong, Anything that throws a wrench in BatCat is always welcomed in my opinion.