When talking about a maternal unbroken linage and a paternal unbroken linage, is the meaning that Mitochondrial Eve had a linage of only women reaching every person today, and the same with Y-chromosomal Adam and men?
Kind of, yes. The concept behind Mitochondrial Eve and Y-Chromosomal Adam has to due with cell replication and basic genetics.
The Mitochondria ("Power Generators" of the cell) are directly inherited from mother to daughter cell. That is, new cells do not start with 0 and are forced to make them de novo. Some Mitochondria are replicated before division and some are 'given' to the daughter cell to start with.
Because of the unique way that Mitochondria are inherited, when it comes to creating new humans it means that only the Mother's Mitochondria are given to any children she gives birth to regardless of the sex of the child. So I, being male, have my Mother's Mitochondria - which the same as my Grandmother's - and the same as my Great-Grandmother's - etc. etc. etc. until you can trace that line back to "Mitochondrial Evel" whose Mitochondria are the origin for all humans on the planet and are inherited Maternally.
In that sense it is an unbroken lineage of genetic information as inherited from mothers, though not exclusive to women.
"Y-Chromosomal Adam" is a similar concept - Women do not have a "Y"-Chromosome, so every Y-Chromosome men have today had to originate from somewhere. It's trickier to do than tracing Mitochondrial DNA because while Mitochondria are inherited exclusively from mother to child, there could have been multiple Fathers to Mitochondrial Eve's children and other similar complications that muck things up.
How is the specific "identity" or time period of those persons is known, assuming that researchers didn't check the Mitochondrial DNA or Y-chromosome of every single living human being today?
For "Y-Chromosome Adam" it's still more or less in the data-gathering phase as far as I was last aware. It's not as concrete as "Mitochondrial Eve", which has a very solid body of work supporting her.
The "identity" is known because the more in common a person's genes have with the genes of another, the more closely they are related. The researchers took a lot of genes from very divergent populations, and compared them. The researchers were then able to tell which portions were common among all of the Mitochondrial DNA, which had changed the most, and which had changed the least from each other. Using that data, you can construct a rough image of what the originator (or LCA - Last Common Ancestor) had for their own genetic makeup.
You can construe a time period because portions of your DNA (either Mitochondrial or Nuclear) acquire/tolerate mutations at an expected rate - say 1 mutation every 1.5 Million Years. So as long as you have a rough idea of what your LCA's genes looked like (see above), then a modern day sample with 5 mutations is 7.5 M.Y. away from the LCA.
how can they know for sure that there aren't some tribes in the Amazons, which have a direct female linage to Mitochondrial Eve's sister?
They didn't when they started testing. That's part of the fun of the project.
The "Out of Africa Hypothesis" rose in prominence partially due to the findings of the research. It basically posits that the ancestors of humans evolved once in Africa, and then spread from Africa to the rest of the world. Because the Mitochondrial Eve research concludes that "Eve" was around before we spread throughout the rest of the world, it implies that every ancestor already had Eve's legacy when they started migrating into Europe, Asia, the Philippines, etc. Thus every human on Earth, no matter where they are today, shares traits with Mitochondrial Eve.
For the Amazonian tribes, they would have inherited it from their ancestors in North America, which were ancestors of tribes in Asia who crossed an ice bridge into North America, and the tribes in Asia came out of Africa.