First of all the process does not state that it cannot accept its own product as an input. In order for the product not to be a valid input, it would need to specifically exempt it, as it also specifies both that the process works on any poison that comes in doses (not exempted, of course) and that the end product is such a poison.
Next, the ability does state that you need two doses of the same poison. While a case could be made that the base poison and its extra-strong form are the same poison, it seems pretty clear from context and the workings of similar rules that they should not be. So, particularly given your balance concerns, of the two equally RAW interpretations the one differentiating between concentrated and non-concentrated poisons is probably the better choice, especially if Paizo ever publishes a poison variant of the contagion spell.
With outside assistance, there is no maximum limit to the concentration of poison, though the resources required to achieve each level past the soft cap are exponential in growth. Without outside assistance the total time required in minutes is \$2^n - 1\$ where \$n\$ is the desired concentration level of the poison, and the cap is hit when the difference between a level and the previous level is more than 50-60 minutes (depending on GM ruling as to when the dose is used, RAW seems to slightly favor the shorter time limit. We will see later it turns out not to matter, though). This makes the rule:
$$(2^n-1)-(2^{n-1}-1) \le 50$$
or
$$2^n-2^{n-1} \le 50$$
or
$$2^{n-1} \le 50$$
so the cap is at 6, since the 7th level would require 64 minutes of preparation to make, and the 6th level only requires 32. Conveniently, this eliminates any need to resolve our previous ambiguity as to when, exactly, a dose is used.
The 6th level of concentration results in a poison that costs 64 times as much, takes 63 minutes to make and is spoiled by even the slightest interruption, lasts for only an hour itself, and provides a 1039% increase to duration (~11× as long) and +12 to the save DC. This is a fairly expensive option for the meager power it provides, but certainly goes a long way towards making poisons of the less combat oriented type much more viable when used (since you will generally have time to prepare them safely).
Ok, so, for curiosity, I've been trying to optimize the solo crafting order to maximize the highest level of potion created. This is pretty much just a math problem; don't continue reading this unless you are interested:
My best so far is:
first hour: get 60 level 1 poisons from 120 level 0 poisons (regular doses)
second hour: get 30 level 2 poisons from 60 level 1 poisons; get 30 level 1 poisons
third hour: get 16 level 3 poisons from 32 level 2 poisons (two of which were crafted at the end of the cycle using produced level 1 poisons), get 30 level 2 poisons from 60 level one poisons (leaving us with 28 left for next time), get 14 level 1 poisons
fourth hour: get 8 level 4 poisons from 16 level 3 poisons, get 14 level 3 poisons, 8 level 2 (one produced from lv 1 poison from this turn), 30 level 1 (28 at end of turn)
fifth hour: 4 level 5 poisons, 8 level 4 poisons (1 produced from threes made this turn), 4 level 3 poisons (leaving 2 for next turn), 15 level 2 poisons, 28 level 1 poisons
sixth hour: 2 level 6 poisons, 4 level 5 poisons, 2 level 4 (1 using poisons produced this turn), 8 level 3 (1 using poisons produced this turn, leaving 6 at end of turn), 14 level 2 (leaving 12 at end of turn), 30 level 1 poisons
seventh hour: 2 level 7 poisons (1 using poisons produced this turn), 2 level 6 poisons (0 at end of turn), 2 level 5 poison, 3 level 4 poisons (1 at end of turn), 6 level 3, 15 level 2, 30 level 1 poisons
eighth hour: 1 level 8 poison, 0 level 7 poison, 1 level 6 poison, 1 level 5 poison (from 1 level 4 produced this turn), 3 level 4 poisons (2 at end of turn), 8 level 3 (1 from 1 level 2 produced this turn, 15 level 2 (14 at end of turn), 30 level 1 poisons.
The Rules Are... Ambiguous
The Universal Monster Rules regarding poison say that
... The saving throw to resist a poison is usually a Fort save (DC 10 + 1/2 poisoning creature's racial HD + creature's Con modifier; the exact DC is given in the creature's descriptive text).
The only ambiguous part of that rule regarding the centipede as a normal creature is the 'usually a Fort save' segment; the rest specifies how a poison's save DC is calculated, and that it will be calculated in the creature's descriptive text. Recalculating on a change of racial HD or of Con is the correct reading of the rule.
The rules are silent, however, as to whether the effective HD for effects are racial or not. One argument in favor is that the familiar has only one type of HD, and so they would be 'effective racial HD.' The argument against it is that the rules don't say that they are.
Both rulings have merit; the permissive ruling helps the familiar remain relevant as the master's level increases, and the restrictive ruling helps keep 'the rules don't say I can't' from being a useful argument.
Further Analysis
The clauses from the familiars section regarding HD is dichotomous:
... It retains the appearance, Hit Dice...
vs
Hit Dice: For the purpose of effects related to number of Hit Dice, use the master's character level or the familiar's normal HD total, whichever is higher.
The second clause would override the first, depending on the definition of 'effects'. Since there is none in the glossary, extrapolation is necessary.
Within the glossary alone, there are hundreds of instances of the word 'effect' or 'effects'. Looking at the Afflictions Section, afflictions, including poisons, appear to have effects.
Thus the effects of poison that are related to HD should be calculated from the master's HD, when higher.
It's still ambiguous though. If they'd just define what an 'effect' is, then these questions would be a cakewalk.
Best Answer
The Drow would still deal poison damage after a minute
Creature statblocks only do what they say they do. In the statblock for a Drow, they are listed as having a crossbow attack that targets must save against or be poisoned. There's no mention of poison needing to be actively applied to maintain this effect, or of the Drow even having a supply of poisons on-hand.
DMs can adjudicate the application of a Drow's poison, as a houserule/rule of fun
Personally, as DM, if I knew a creature's poison attacks were probably caused by a manually applied poison, I'd probably narrate that the poison is applied on either the first round of combat, or in an ambush scenario (which is traditionally where Drow show up in combat) they'd have applied it right before combat began. Alternatively, if the PCs were to get the drop on the Drow, it might be that the Drow have to choose between fighting without poison, or wasting their first turn applying poison.
One thing to note, however, is that the one minute limit on the poison duration only applies to Player Characters; it is not required to apply to NPCs of any kind. And as DM, you're not required to apply any kind of adjudication on this topic at all: if a statblock says the creature deals poison damage, then they deal poison damage, with no requirement to take into consideration other factors like time or circumstances.