Background
On initial creation I usually use the optional point buy system. Which allows 27 points to be distributed across the 6 abilities, with higher scores costing more points than lower scores (PHB pg 13). Later in the customization section (PHB pg 165) it states that an ability bonus (occurring at levels 4, 8, 12, 16 19) can be replaced by a feat.
Question
Is there a way to replace some ability points with a feat upon character creation?
First is there a dnd-5e official way in some hidden an addendum somewhere or in past version that would translate? If not what would be a fair point cost for a home brew rule? I was thinking 5 points. Which could effectively cost 2 ability score.
Scenarios where this might be useful.
I am creating a bard with a custom background of "Diplomat". Based loosely on the noble and entertainer backgrounds. The background would allow several additional languages but official rules only allow 3 perks to a background. So the Linguist feat might help get the build I'm looking for.
Also a barbarian could use the Tavern Brawler feat and almost break even with a few extra skills with improvised weapons supporting a background in urchin or sailor. (The point cost of 5 would insure it's not completely free as Tavern Brawler gives +1 to con and str)
Starting with Heavy Armor Master as a Paladin or Cleric by sacrificing a couple points in dex or int for a better tank build on start.
Clarification
*By points I mean of the 27. And score would be the value 8-15 that is purchased. I think this is how the book uses the terms.
Best Answer
Yes, but only for a human, and only at the DM's allowance.
The only rules-based way to grab a feat at character creation is by using the "Variant Human Traits," found on PHB p.31. There you sacrifice four +1 to ability scores to gain a skill and a feat. It's not exactly sacrificing points from your point-buy, but the effect's the same.
Past versions have little bearing: they're different games, and while they're great to reference for source material, ideas, content-lineage, balance is the worst place to go looking at a different game.
From this edition we can see that your proposed homebrew solution would be rather unbalanced: it costs much less in your scheme than in the published version. (Which I presumptively consider balanced.)