Your class and all starting class features, such as skill proficiencies, are chosen before backgrounds — see PHB pages 11–15 for the order of operations. So yes, you can select overlapping skills for your barbarian before taking the Outlander background, then get to choose any two skills instead.
A minor reason you might not want to do this is because the trick locks you into choosing Outlander / not choosing any other background.
It leads to "All or Nothing" skill development
This is going to cause exactly the problem that the 'bounded accuracy' philosophy of 5th edition was designed to address: All skill checks become either trivially easy for experts, or impossible for everyone else.
That is to say, the DM either sets skill DCs low enough for everyone to have a fair shot at making it, in which case the expert almost can't fail (and in that case, why would the non-experts even try?), or the DM sets the DCs high enough for the expert to be challenged, in which case everyone else can't possibly do it (so again, why even roll?).
This leads inevitably to a situation where there isn't much point to basic proficiency; everyone wants to dump all their advancement into a few skills in order to be "the one" for those specific skills, and everyone else avoids those skills like the plague because they know they can't be good at it.
Yes, your idea would allow characters to become more highly specialized -- but that actually isn't a desirable outcome. It isn't fun for one character to be the God Of Investigation while everyone else has to just wait for that one character to handle all the Investigation rolls. Maybe that one guy gets a moment of power-fantasy gloating -- "Ha, nothing escapes my eye!" -- but the rest of the table is effectively being told, "You cannot contribute in this situation." And that's not fun at all.
As a side problem, your concept virtually removes the role of ability scores in being good at a skill. At some point (probably around level 10), the value of the trained skill bonus completely overshadows the ability score's contribution, to the point that it almost doesn't matter if your character is agile or not, charismatic or not, all that matters is what skills you trained up.
Best Answer
No, proficiency bonuses never stack
You can only ever receive a proficiency bonus once regardless of how many sources grant you proficiency in it. When picking skill and tool proficiencies granted by a background or class, Players should try to avoid overlapping proficiencies as the benefits are nil
However a proficiency bonus may be increased or decreased based on circumstances
DMs can use this as a way to give a bonus to skill checks based on good roleplay and/or planning.
The Rogue's Expertise feature is an exception
While not allowing proficiency bonuses to be counted more than once, expertise allows a Rogue to double the benefit of their proficiency bonus to two skills (or thieves' tools) at level 1 and again at level 6.
Backgrounds granting a redundant proficiency let you pick a different proficiency instead
In the Backgrounds section of the Basic Rules & PHB, the following rule is listed:
(Note that, while unclear, this rule may only apply to background proficiencies.)
Custom backgrounds are also an option available to players to avoid duplicate proficiencies due to backgrounds.