Officially there is not explanation about how they are even connected let alone how they left the Milk Way for the Ida galaxy and why the Asgardians didn't knew about them before. For the most part everything about the Milk Way Replicators are just but pure speculation by fans. Only the Asurans had a background explained. Reese's appearance made no advancement in explaining who was responsible for the Replicators, they just added a new robot to be the source of the other robots but who created the first robot, Reese, still remains unknown.
But I've always wondered how it would be possible to explain their similarities and the fact the Ancients had an specific weapon to destroy Replicator blocks.
This is my idea in how it could have happened in any way this was explained in any type of media;
I think Reese was created by an Ancient scientist who lived in the Milk Way before the plague. Much like we saw in other instances, the Ancients had worked with many different researches over their time in our galaxy and some of them didn't function properly or were at very early stages of development such as their time technology as seen in Window of Opportunity.
For me, Reese was created with the purpose of researching in how to build an advanced artificial being. The Replicators could have been intentionally part of her project, a robot who could create new robots such as some algorithms we have in real life computing which can evolve and develop new and better algorithms than the original ones, such as Evolutionary Algorithms and Genetic Algorithms or they could have been an unintentional byproduct of her design, since she was clearly created to act and think like an organic being.
She then lost the control of her Replicators and they made a mess in the city but eventually the Ancients or the Ancient responsible for Reese created the disruptor or an predecessor of this technology and ended the Replicator uprising. This would also explain why they found Replicator blocks lying around the planet.
Reese's father then decided to put her to sleep, because he/she didn't wanted to lose the project. It wouldn't be the first time an Ancient hid a project which other Ancients decisively disagreed with the progress, such as the Time Jumper by Janus.
It's also possible those hypothetical Ancients living on Reese's planet had achieved a way to input a code into the hive-mind of the Replicators making them, the Ancients, not a target as they did with the Asurans.
During the Ancient-Wraith war, it's possible they decided to revive the project, already knowing how to stop the Asurans if needed and also how to control them direct on their programming, seemed to be a nice idea. Nonetheless, again the Replicators started to generating trouble when they adapted and decide to kill humans in order to get rid of the food source of the Wraiths and were shutdown again as we know.
The Ida galaxy Replicators could have travelled with an Ancient ship. If the idea of the Ancients from Reese's planet had achieved a way of making themselves invunerable to the Replicators, a single entity could have survived and travelled with the Ancients during one of their trips to Ida, which probably happened a lot since they and the Asgard were in an Alliance. This would explain how a Replicator could travel in an Ancient ship and not attacked it, since it wouldn't have a taste for anything Ancient because of their programming. It is also possible some Ancient had decided to bring the project to Ida galaxy in order to continue it without the others knowing.
I like to think that's what happened, it makes sense in my head. I know probably the Asurans were added because they wanted something like the Replicators back and didn't care much about the connections.