Yes, there's nothing (in the rules, anyway) preventing multiple Eldritch Knights from bonding with a single weapon.
Once you have bonded a weapon to yourself, you can’t be disarmed of that weapon unless you are incapacitated. If it is on the same plane of existence, you can summon that weapon as a bonus action on your turn, causing it to teleport instantly to your hand.
You can have up to two bonded weapons, but can summon only one at a time with your bonus action. If you attempt to bond with a third weapon, you must break the bond with one of the other two.
However, if the weapon is a magic weapon that requires attunement, only one person can be attuned to it.
As for breaking someone else's bond with a weapon, this is probably only possible with a well-worded Wish. Dispel Magic is the go-to spell for getting rid of annoying magic, but it only works on ongoing spells, which the bond clearly isn't:
At 3rd level, you learn a ritual that creates a magical bond between yourself and one weapon. You perform the ritual over the course of 1 hour, which can be done during a short rest. The weapon must be within your reach throughout the ritual, at the conclusion of which you touch the weapon and forge the bond.
Similarly, the bond isn't a curse, so Remove Curse won't help. You can render the bond useless by being on a different plane of existence to the Eldritch Knight, or being in an Antimagic Field. On the bright side, depending on how your DM interprets the phrase "you can't be disarmed of that weapon", it might be impossible for the other Eldritch Knight to summon it while you're holding it.
The co-operating knights can take turns to use it, but again, depending on how your DM interprets the phrase "you can't be disarmed of that weapon", they might need to drop it at the end of their turns so that it can be summoned by the other knight. Even if they don't have to do this, they're still going to have problems making opportunity attacks, since only one of them will actually be holding a weapon at any given time.
No, it is not magical
Jeremy Crawford clarified this point on Twitter, using Sage Advice to support it. It is "background magic" similar to the breath weapon of a dragon.
Grant Myers @realgrantmyers
@JeremyECrawford ki is described as magical, but nothing in stunning strike says that it's magical. Can you confirm that it is not please?
Jeremy Crawford @JeremyECrawford
Neither the Ki feature nor the Stunning Strike feature (PH, 78 & 79) is defined as magical for game purposes. #DnD
Joe Lastowski @JoeLastowski
Replying to @JeremyECrawford
Except in the section called "The Magic of Ki" at the start of the Monk description.
Jeremy Crawford @JeremyECrawford
That is an example of the background magic I talked about in Sage Advice. Look for "Is the breath weapon of a dragon magical?"
Therefore, just as a dragon's breath weapon is not considered magical, a monk's ki is also not considered magical. They all work in an antimagic field.
However, Ki-Empowered Strikes are specifically magical
The feature says:
Starting at 6th level, your unarmed strikes count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
This is also supported by Jeremy Crawford's tweet.
Jeremy Crawford @JeremyECrawford
The Ki-Empowered Strikes feature says a monk's unarmed strikes count as magical. That magic is suppressed in an antimagic field. #DnD
Best Answer
Blood Hunters only seem to form short-term "bonds" with their weapons
The description of the Blood Hunter class does indeed say:
This is referring to their Crimson Rite class feature, where the Blood Hunter sacrifices some hit points in order to power up a weapon they're wielding. However, these "bonds" are quite temporary and not permanent, as the rite describes (emphasis mine):
So the bond they form only lasts for as long as they're holding on to the weapon, and even then only until the next time the character rests. It doesn't really represent a long term bond of the kind you're imagining. However there are other options in the game which could fit with this kind of concept.
Pact of the Blade Warlock
A class which does have that kind of association with a weapon is a Warlock choosing the Pact of the Blade Pact Boon. They have the ability to summon a "pact weapon" and can even perform a special ritual to bond themselves to a specific magical weapon, which might be closer to what you're thinking of! If you're a Critical Role fan, you might recognise this as being the class and option that Fjord has. A Warlock always has a mysterious patron who grants them their abilities and knowledge, but it could easily be the case for a Pact of the Blade Warlock that their patron communicates with them through their pact weapon.
Sentient Magical Weapon
Alternatively, it is possible for magic items to be sentient individuals in their own right, as described in the Sentient Magic Item rules. You could easily imagine an ancient magical sword with its own personality, acting as a mentor to the young adventurer who found it. It'd be very unusual for a low-level adventurer to have that kind of powerful magic item in most circumstances, but if your DM really likes the idea and thinks it could be interesting he might be willing to work something out with you - if not starting out, maybe planning for your character to find such a weapon as you play so you can explore that kind of story.