Ok, say I have Manaroo out and she has a target lock (blue side) on her ship, IE she is targeting something. Further, at the start of combat I give that target lock to another ship per her ability. Assume that the new ship also has a blue side target lock on their ship from their action. Can that ship now have 2 target locks?

Remember:

1 was an action by the ship itself

1 was assigned per another ships ability

Unless the other ship has a Weapons Engineer on board, it cannot hold two target locks at the same time, so one must be discarded. How they got them is immaterial.

Page 3 of the Rules Reference:

"A ship can maintain one target lock. If a ship acquires a new target lock, it must remove its old target lock."

I believe there is one more exception to only being able to have on ship "locked on" but generally ships can only have a TL on one other ship at any given time. If they get a new one they are required to give up the old one.

Think of it this way. TLs generally stick around from turn to turn unless you use them but you can only have one at any given time. Focus and Evade tokens are generally discarded at the end of the round but there is no hard limit how many a ship could acquire during the round

Ok, say I have Manaroo out and she has a target lock (blue side) on her ship, IE she is targeting something. Further, at the start of combat I give that target lock to another ship per her ability. Assume that the new ship also has a blue side target lock on their ship from their action. Can that ship now have 2 target locks?

Remember:

1 was an action by the ship itself

1 was assigned per another ships ability

No.

You can only ever hold one TL unless you have a Weapons Engineer or are Redline. If you get another blue token over and above what you can hold, however you get it, you discard blue tokens and their associated red ones until you reach your maximum.

Ok, say I have Manaroo out and she has a target lock (blue side) on her ship, IE she is targeting something. Further, at the start of combat I give that target lock to another ship per her ability. Assume that the new ship also has a blue side target lock on their ship from their action. Can that ship now have 2 target locks?

Remember:

1 was an action by the ship itself

1 was assigned per another ships ability

No.

You can only ever hold one TL unless you have a Weapons Engineer or are Redline. If you get another blue token over and above what you can hold, however you get it, you discard blue tokens and their associated red ones until you reach your maximum.

According to the rules quote above, you have to keep the new one. So if Manaroo passes to clear an enemy red target lock, the receiving ship has to also take whatever Manaroo's blue target lock is, even if that's not as good as their old target lock.

Is "acquire" different than "assign" in this context of removing your old target lock when you "acquire" a new on?

It is for some abilities like Jendon where the ship you assign the target lock to does not have to meet the normal range requirements for acquiring a target lock.

There's the line about only maintaining a single lock at a time so it is mostly moot.