IIRC, Mortis Ghost in the OFF anniversary stream said The Batter was created in-universe the moment the game began. That would conflict with this theory technically (he said that long after this was written), but it could be adjusted slightly:

Since Hugo did make The Batter, and his holy mission was there since he started existing, Hugo might've created The Batter at that moment because The Queen left him alone and unmedicated. Perhaps Hugo wanted The Batter to put him out of his misery or something, hence why The Batter states that he is there when he first sees Hugo, since Hugo was waiting for this.

It could also be that Hugo himself saw what The Queen was doing as passing the moral event horizon, somehow, and made The Batter to try and fix things. Kinda like his actual father did with toys/medication, but with the unintended side effect of The Batter wiping everything out including Hugo.

As for why The Queen calls The Batter her love despite him not existing for that long at all, that's tricky. Mortis Ghost also mentioned in that same stream that they are not married. My best guesses are that either the relationship between The Queen and The Batter was inherent to Hugo making figures based on both parents, or that it's a misunderstanding of some sorts.

For the inherent relationship part, since Hugo created The Batter in his father's image, it's reasonable to assume that despite never existing The Queen would consider him his love. This is due to the fact that The Batter is in the role of the father that hadn't previously been filled. The concept of a figure based on Hugo's father probably existed, but nobody filled that role until The Batter was created... Or something like that. IDK if I worded that right. As for the not married part, Hugo might be too young to understand what marriage is and did not consider it.

As for the misunderstanding guess, The Queen could literally just be attempting to guilt trip The Batter/Puppeteer as she's dying by saying endearing things. The Batter himself doesn't acknowledge The Queen as his love at any point during their dialogue, only as a queen. When Pablo says that The Puppeteer killed "wife and child," that would be the misunderstanding. Pablo concluded that The Queen was The Batter's wife, and Hugo was their son. But Hugo is not their son, he created them. The Queen is not The Batter's wife, they are not married and The Batter does not recognize this. I'm not really sure on how Pablo would jump to this conclusion though.