No, it's not lore-breaking, that would require an internal contradiction. It's just unrealistic. Then again, so are the human-looking aliens who want to have sex with you.
It is an internal contradiction. The armor in the ME universe is supposedly a reasonably effective short term defense against the firearms used in said universe (otherwise no one would wear it), which produce many orders of magnitude more energy than a sword. If it can block or mitigate the multiple thousands to perhaps several hundred thousands of joules that these weapons are putting out, than a sword with its maximum 60-130J shouldn't even make a scratch.
It does, though. Therein lies the contradiction, and the sword is in fact more powerful than most if not all of said firearms. No justification is given for this apart from "monomolecular blade". The blade clearly isn't a single molecule thick across the full cross section as it is easily visible to the naked eye, so this can only refer to the edge. Edge sharpness has virtually nothing to do with cutting ability in comparison to profile. It could be a "monoatomic" or "mononucleon" or whatever other nonsense technobabble they can come up with and it still wouldn't matter, because the flat of the blade and thus edge geometry isn't getting any narrower and the energy requirements to overcome drag are thus the same.
I forgot to even mention that with how easy these blades are to break (i.e. a simple impact from Shepard's backhand or a shot from in universe firearms) they shouldn't be able to cut a wet paper bag without shattering into a million pieces, let alone hold together through the couple of dozen kilajoules of force required to get through armor.
You have no way of saying that such aliens are realistic or unrealistic, given that your sample size of planets with evolutionary histories is limited to 1. There are even theories such as convergent evolution that presuppose that the process is rather more limited than one would think. I'd reasonably say that given the sheer size of the universe, your assertion has a non zero probablity of being incorrect, and is therefore more likely than that of a human being able to swing a sword with several thousand joules of force.. It is literally impossible. Your muscles can't generate even a fraction of that, and your bones can't come close to coping with such forces. Further there is no existing metal with the tensile strength necessary to hold itself together even if you could.
EDIT:
Your reasoning is based on the assumption that the light ceramic armors of Mass Effect are equivalent or better than current metal plate armor. Considering that the Banshee's bare hands are capable of penetrating it, this assumption may be false.
There is no "current metal plate armour" because ceramics have been proven superior in every conceivable way, being both lighter and more dense, thus stronger. The numbers I ran were for totally not "current" wrought iron, which is a fairly weak, brittle metal from the middle ages. If a sword has no hope of even coming close to defeating that (and it doesn't), than its effects on the former would be utterly laughable.
Would you like me to run the numbers as to what that armor in its modern form can stop? Let's start with a simple type II Kevlar vest (no ceramic plates), which is rated to stop modern handgun rounds such as the 9mm (600-700J), .45 ACP (700-850J) and even .357 Magnum revolver (1000-1100J). You'll note all these numbers are many times higher than the sword, and they are being exerted over a smaller surface area as well.
When you start getting into hard armors such as used by any modern military force or even police special units, it gets even more absurd to think about. Type III with its conditioned ceramic plates can stop a 7.62mm NATO rifle round (3,000-3,500J). Type IV (Such as the US military's MTV armor) with its ESAPI plates of boron carbide and backed by Spectra (a type of ballistic polyethylene that is 40% stronger than Kevlar) will stop .30-06 or 7.62x54R specialized armor piercing rounds (4000+J). Furthermore a sword, due to its much greater surface area that is impacting the plate or vest than a bullet, would need to generate several times those amounts of energy to defeat those armors. Keep in mind it was 21,000J for the wrought iron, and you start to see just how absurd the notion is that these swords are anything more than hilariously cumbersome paperweights.
Assuming that armor technology hasn't somehow regressed, ME ceramic armor should be at least as capable as the modern version. However, there are indications that ME universe firearms produce far, far more energy than modern ones do, meaning that the armor is similarly orders of magnitude more effective in order to reliably stop them.
Is it really more technobabble gibberish than molecular nodes of ore in a person's brain giving them telekenisis via dark energy manipulation?
We've already broken the laws of physics in the name of rule of cool to give us a mage class. I see no problem in doing it to give us some melee classes.
Given the choice I would remove pure magic like biotics from the setting as well, but that battle is probably not worth fighting given that they have been central from the start. Besides, at least with telekinesis I can actually see some potential utility. That doesn't exist with a sword, which are inferior to firearms in every way.
Idiotic Metal Gear mall ninja crap only started showing up in the latest installment, though. No need to keep it around. If you want to play with swords, Bioware makes a game for that IIRC. "Variety" is a silly argument that would equally justify putting sniper railguns in Dragon Age. The game is a science fiction TPS. There's no need to have an extensive melee system at all, let alone dedicated weapons that make no logical sense in universe given the type of combat and other technologies utilized.
Nonsense is not a justification for more nonsense. If anything, it is the opposite in science fiction. You invent a few contrivances (i.e eezo and its effects, including FTL travel) and you go from there. You don't continually add yet more obvious contrivances that contradict previously established lore just because "cool" unless you want to simply make a space fantasy.