Or should just the initial shot be aimed and the placement of the second shot near the first shot just developed over time with practice and muscle memory?
Double Tap is no longer a real practice in any modern day military (or really ever was... officially). The Army today trains on controlled pairs. This is where you practice reflexive firing (point shoot method) also known as CQM/CQB by military/operators.
The actual firing method of control pairs has to do with close quarters marksmanship (CQM) training. The shooter acquires the target by their eyes only and with the reflexive training brings the firearm muzzle up to target while still keeping eyes on the target. The weapon is always shouldered throughout the entire process as this is called the "low ready" and the only accepted position for urban/confined combat rifle techniques. Pistols use the same principle but with elbows slightly cocked/bent, close into the body, and the combat pistol technique implemented (push pull method of firing acquisition). This can be done by shoot, turn and shoot, side step shoot, shoot walk, etc.
Your sight picture will determine the speed and acceleration of the controlled pairs. They are not just "bang bang" but precise and deliberate. When the sight post is on target the shooter will instinctively and quickly squeeze off the first shot, watching the target until the front sight post comes back on target where the next round is quickly and deliberately squeezed off. It will resemble a circular motion when done correctly with very little disruption or pause as the recoil is seen through the front sight.
The weapon muzzle is then dropped slightly into the "Low Ready" while the target is quickly determined and/or a new target is acquired by the shooter. If the target did not change shape that you just engaged the CQM technique is to continue until it does or it is determined to be a non-threat or a greater-threat is identified.
Yes it is a somewhat long and lengthy process as far as training is concerned BUT once learned it is very valuable to the aggressor. Note I said AGRESSOR because CQM/CQB is to be used in an offensive/movement type manor as defensive strategies are ambush and that is not what the controlled pair is designed for (or what people call double tap).
And on a completely separate note: if you want any credibility with the Para-military/operator/military/other-than-recognized US hired forces, refrain from calling or even using the term "double tap". Use the term CQM for the shooting style, and controlled pair when describing your shooting technique (though everyone uses it anyway during CQM so it goes without saying).
Hope this helps.
Good Luck and be safe!