Lotta people seem unsure of what happened in the show, so I'll clear some things up.
The twins' mother did, indeed, love her children. She had names for them. She tried to tell Klaus Poppe their names while she was still pregnant, but he wasn't inclined to pay any attention to such trifles as "names" and "individual identity." The names she had chosen were meaningless, since the children would soon be in the custody of the Czech secret police, to be shaped into the leaders of the glorious future of Communist Czechoslovakia.
It's heavily implied (but not shown) that, once the twins were born, the mother snuck off the grid and hid out in that tiny, crappy little apartment above the Three Frogs. Part of the deception was to dress both twins the same, so that the world outside would believe she only had one child, and of course this isn't the woman we're looking for, because she has only one child instead of twins! But it didn't work forever, of course.
When Poppe came to take one of the children away in 1981, the mother vehemently defended her children. But there were 3 dudes with no morals vs. her. And so, the mother's advent as "the true monster" came when she made the bitter, calculated choice to hand Anna/Nina over to Poppe, in the hopes that Johan would foster the same hatred towards Poppe that she felt and exact revenge when he came of age. Which was true, but of course, had the unintended side-effect of obliterating any ounce of love and trust Johan may have had for his mother.
At some point in the next few days, the mother was abducted and taken away, leaving Johan behind.
Then we get the stuff we already know well: Anna/Nina is taken to the Red Rose Mansion, she is imprisoned in the padded room, she is brought out to witness the massacre, and Klaus Poppe (with his sudden, last minute reemergence of a long-suppressed conscience) swiftly tries his damnedest to deprogram the child so that she doesn't turn into the amoral monster she was being groomed to become. But unfortunately, the damage was already done to Johan, who could never trust anyone or forgive anyone ever again after the incident. As we see in the last episode, the very thing that drove Johan to evil his entire life was the question of, "Which one of us was the unwanted child?" He had no knowledge of any revenge plan or the circumstances of his birth or anything about the secret police--all he knew was that his mother, the person he trusted the most, willingly gave one of them away to face the horrors of the Red Rose Mansion. That kind of thing fucks you up for life, when you're only 5 years old. And because Poppe deprogrammed Anna/Nina, she didn't let her experiences warp her, but Johan had no such luxury.
And so the twins' mother lived out her life, eventually ending up in that French Catholic institution, where she must have just assumed the worst happened to her children and they were left to die without her there for them. She bursts into tears when she learns that they're alive and well, and she laments that she had forgotten the names she had given them up until now. She gives them to Tenma, in the hopes that maybe, after everything, she could at least provide one last piece of closure to her estranged children.
As for why they "had no names," they clearly must have while they were in their mother's care. But because Johan felt betrayed by their mother, he abandoned the names they had been given and began patterning his life around "The Monster Without a Name." The dub kinda flubs this plot point by having young Anna/Nina call him Johan instead of "brother," but that's hardly the biggest sin Monster's dub commits. When young Anna/Nina asks her brother to call her by her name, he refuses, saying that they don't have names ("anymore" being implied). Johan had fully committed to the idea that the two of them would wander the world without names--without identities--and that the only thing they would ever need was each other.
You know what else doesn't help? Their savior and their brand-new parental figure, General Wolf. Takes the twins in and saves their lives from death by starvation, and then...whoops! Time to carelessly send them to different orphanages! Johan, in particular, being sent to an orphanage whose primary mission was to brainwash children into being the "future socialist elite." Meaning, stripped of their individuality and their identity and (in a...strange contradiction?) pitted against one another to establish dominance. Once again, a parental figure Johan might have trusted completely and utterly betrayed him. C'est la vie...
This leads into the question of, "Well, why da fuqq was Johan killing all those couples who took them in?" The answer being grimly simple: So that Johan (and his sister) would never have to feel the pain of parental betrayal ever again. Everything was going to be on Johan's terms. They receive their care because he wills it. If he feels like his sister is getting too attached, too emotionally-invested, it was time for the parents to go. On to the next. With the deaths of the Lieberts, Johan is finally confronted by Anna/Nina, and he gives her the choice--to forgive him, or to punish him. Johan was really hoping for the former, hoping that his sister would remember their mother's betrayal and understand him and his behavior, but of course, that's not what happened. The two had already drifted so far apart in their worldviews. And any hope that Johan might have been deprogrammed was gone forever...until, possibly, maybe, hopefully, his being saved by Tenma for a second time 12 years later.
I could go on and on, but that's everything for now.