Does a Character have to spend a Maneuver every round to Aim at the same target. Or, if they Aimed one round, and haven't lost "sight" of their target or moved themselves, Does the previous Aim remain in effect and they can use a maneuver to get the double Aim bonus the next round?

Aim provides 1 boost die per Aim maneuver (maximum of 2) to your next combat check if you do nothing else in between. Important part there is "next"

So yes you need to re-aim after each attack.

And as to the "why" once again it comes down to the fact that both movement and time are abstract. Just as we have all discussed before an attack action isn't a single firing of a gun or swing of a Lightsaber. Aiming isn't getting the target in your sights once, it represents dedicating your time to keeping them in your sights while they move around the battle field, and also not being distracted by the action around you. If you're not dedicating time to keeping them in your sights then your not getting the benefits of aiming.

Important addendum: you can aim one round (and do nothing) and aim a second round then shoot and get both blue dice.

Does a Character have to spend a Maneuver every round to Aim at the same target. Or, if they Aimed one round, and haven't lost "sight" of their target or moved themselves, Does the previous Aim remain in effect and they can use a maneuver to get the double Aim bonus the next round?

Everyone answered your first question nicely, but to answer your second question, yes. The aim maneuver continues to the next round. Keep in mind though, the narrative nature of the system makes it possible for something to break your aim. The most common ways are an attacker using advantage to do so, if your character takes wounds, and if you either move or get moved. The easiest way to get double aim is to just take the strain for the extra maneuver. Aim, Aim (1 strain), Attack.

Also, please not that you don't actually aim at a target. You just aim. The only time you have to specify what you are aiming at is when targeting a specific carried item or part of the target. This is useful to remember because a target running out of sight doesn't ruin your Aim benefit on your next attack. You can just use the benefit on a check vs someone else. Expect some limit to that though. Most GMs won't be so keen to give the benefit of aiming if for example you are aiming with your blaster rifle out then decide instead to rifle butt the guy who just showed up between rounds.

Also, also, lots of people forget or just straight don't realize that Aim can be used on ANY combat check (including Force Powers), not just ranged with Ranged weapons. Seems silly, but it's easily overlooked. I even did myself when I was first learning.

This is important:

Important addendum: you can aim one round (and do nothing) and aim a second round then shoot and get both blue dice.

Eg: (written in order of play)

Round 1:

Move behind cover (maneuver)

Shoot (action)

Suffer strain to Aim (2nd maneuver)

End turn

Round 2:

Aim (maneuver)

Shoot with 2 boost extra for double aim (action)

Certain things will break this cycle, normally it would be moving at the beginning of Round 2 before shooting. But other things could cause a "Distraction" such as a failed Fear check, taking damage, the effects of a triumph that cause you to move, certain Force Powers, and I'm sure many other reasons. But this is where GM adjudication is important and every table is going to have a different angle to this.