I played many years of 2e (and BECMI, 1e. 3e. 3.5e, PF, and OSR stuff) and have read the 5e PHB, Basic set, and Hoard of the Dragon Queen and played some short games, so I think I can give some good points of comparison. I'm excluding the Skills & Powers stuff in late 2e from this discussion, that was less like 2e than many other versions of D&D itself.
There are definitely similarities between 2e and 5e — mostly conceptual and "feel" similarities. The individual mechanics are different — pretty much entirely, except for the basic "rolling to hit involves a d20! AC is involved! And there's saving throws of some sort!" kind of things all the editions share; none of the numbers or tables or rolls are identical.
It's almost as if 5e was a do-over of 3e, with the benefit of many hard lessons learned from the 3e → 3.5e → 4e experience, mostly about limiting power inflation but avoiding the risks of trying to mitigate that through making everyone the same. I think it's fair to say that in retrospect 3.5e is a min-maxer's wet dream and dissolves into non-fun at moderately high levels from rocket tag and op disparity; 4e tried to fix that but was in the end unsuccessful because it made everyone vanilla and identical in the process — reminiscent of in WoW, how "At level 1, you kill boars... Now that you're level 70, you can go farm level 70 demon boars!" (See also: Some Thoughts On 2e and 3e's Legacy.)
Power Limiting Through Mechanics
5e uses bounded accuracy and advantage to obtain the same power differential limiting that 2e did with just "fewer bonuses." Everyone's addicted to bonuses now, so you can't not have them, but they take a different tack to try and get a power level more like 2e's. In 2e, for those who aren't grognards, there was a lot less difference in power with level and/or Hit Die. You just couldn't stack many bonuses on top because they weren't available. A level 5 person wasn't infinitely more than level 1 or less than level 10, the curve was less steep. 3.5e easily degenerated into people with +0 bonuses and +30 bonuses and "rocket tag." The 5e mechanics are trying to solve that problem with a different mechanical approach, but with a result that feels like halfway between 2e and 3e power level wise.
You'll hear people with yesteryear's goggles on complain about "all those unbalanced kits in 2e!" If you actually go back and read the kits after playing 3.x+, you'll wonder what the big deal is. Most kits would at most get you a free weapon or nonweapon proficiency. One of the most "unbalanced," the Berserker, lets you get a +1 to hit/+3 to damage by going berserk (raging). An 18 STR gets you all of a +1 to hit/+2 to damage in 2e. Back in the day, combining that into +2/+5 was a HUGE bonus — now, we call that "a first level character's standard attack and damage bonus, if they are poorly optimized."
The 5e class configurability options are like a cross between 2e kits and Paizo's class archetypes — more like the archetypes in how much of the character they swap out, but more like kits in terms of "choose just one, permanently."
Power Limiting Through Randomness
Rolling stats reduced power by reducing optimization. That was stock in 1e/2e and is back in as the default gen method in 5e (things went all point buy in the interim).
The concept of creating and buying "to spec" magic items introduced in 3e has also been removed. It was a lot harder to come up with broken combinations when you couldn't just "sell that thing I just found for 50,000 gp and demand the perfect 50,000 gp item for my build in return." In 3.5e you had a "wealth tax" in that you were expected to have a whole suite of +(level/3) enhancements to everything.
So it feels like 2e, more than it feels like 3e/4e, because you can't custom craft every aspect but have to rely on the campaign and luck to a degree.
Other 2e-Like Stuff
The one thing that really struck me was the art similarity between the 2e PHB and the 5e PHB on the interior art. Especially the full-page color plate pics but also the character treatments really made me think "2e!" while I was reading it. Each edition has had very specific art direction and 5e's looks a lot more like 2e's than any other version.
The skill mechanic has been dialed back from "lots of skills" to more like the 2e NWP mechanic where really you just have a couple, while retaining the 3e style mechanics.
Spells and stuff are just shorter, too. This makes it feel like 2e just because it was a midpoint between the more terse BECMI and the more verbose 3e. (e.g. Knock spell — BECMI: 122 words, 3.5e: 206 words, 5e Basic: 132 words)
And finally — attitude. Mike Mearls and the WotC team have been trying to get everyone to take responsibility for their own game and rules again. The GM guidance in 5e is conceptually similar to the strain in BECMI and 2e per How has D&D's guidance to DMs on when to extrapolate from written rules and when to improvise changed over time? A lot of the confusion over his talk about "living rules" is a straight up conceptual shift - he talks about "living rules," and then people only familiar with 3.5e and/or 4e turn it into "Living Rules(tm)" and debate its exact definition. He's just saying "Goddamnit people loosen up!" In 2e we house-ruled a lot; any 3.5e/4e question about house ruling here is usually answered with a Careful Admonition To Not Do So Lest Ye Upset The Holy Game Balance. They're basically trying to bring some of that old approach back — I guess we'll see if they can change their fanbase as easily as in the past or if that's a Pandora's Box that it's hard to close.
A Bunch Of Non-2e Stuff In 5e
Of course 5e also has a bunch of stuff from 3e — feats, similar core mechanic (THAC0 replaced with the pure d20+bonus vs difficulty), similar way of stating rules, multiclassing, no level limits.
It also has a couple things from 4e — mainly weird proud nails from the "UberBalance" (like the "you have to use your action to make your animal companion attack!" bit). And healing surges transformed into Hit Dice recovery made confusing by sticking the "Hit Dice" term on it. Monsters being not like characters is 4e-ish but also 1e/2e-ish.
It has very little from 1e that wasn't also in 2e; the focus on the "three pillars" of combat/exploration/interaction for example, is familiar from 1e but also in 2e. 1e and 2e were mainly different in that 2e cleaned up the super arcane parts of the rules (to-hit tables) but also cut the super arcane parts of the fluff (most of the DMG, including random harlot tables).
And 5e has some net new stuff not from any previous editions, like the indie-game inspiration and bonds and stuff.
I actually like most of the stuff that 5e has in common with 2e. However, after reading Hoard of the Dragon Queen, I was reminded of the super-railroaded adventures in 2e's reign (stand around while this Forgotten Realms novel unfolds around you!). And that adventure is a railroad from hell. (Many others have talked about that, see here and its linked articles for details.) I hope that's not how that's going to go down. Basically there have been two good runs of adventures in all of D&D history — 1. All the 1e adventures and 2. All the Paizo Adventure Paths. All the other editions have had very occasional gems amidst a sea of dross. This is one part of 2e I hope we don't retread, though the first adventure definitely seems like it.