For me and my definition of old school, no it really really doesn't. Not it the slightest. It seems designed to explicitly be the opposite of what I like about old-school D&D. For me, old-school D&D is a few things that are closely related. Some more necessary than others, but they're all in there.
Weak starting characters. 5E takes the opposite approach. Quite powerful starting characters is the default. High stats, lots of hit points, many skills, lots of powers to pick form, and lots of combat ability for everyone. All casters have infinite cantrips that are as good as or better than most weapons in the game. Various cantrips and 1st-level spells are so good they're broken and these are seen as the default. With the prevalence of optimization, whatever lists of "good - better - best" abilities, feats, skills, spells, maneuvers, etc are widely dispersed and these are seen as the only smart or viable choice.
Zero to hero. 5E takes the opposite approach. Characters start very powerful compared to older editions and their power increases rather drastically from there. In 5E, characters start as heroes become superheroes, demigods, and gods.
Easy character death. 5E takes the opposite approach. There's a wild amount of healing in the game, even from 1st level. Once your hit points are gone you have to take up to 3 more hits or fail three 50% saves before dying...and if even a single hit point is healed...you start that 3-step cycle all over again. So the default healing style in 5E accounts for that and is less than affectionately known as "pop-up healing". So while character death does occasionally happen, it's quite rare in my experience. Even when a bad guy would single out and focus down a specific character healing is so plentiful and overpowered that characters might as well be immortal. Then there's the superhero regeneration of hit points with an 8-hour rest.
Quick character creation. If characters are going to be easily killed off, then character creation should be quick and easy...5E takes the opposite approach. You can streamline character creation up to a point, but even then you're not to the point of old-school D&D speed of character creation. This is also a player base thing. Times have changed and most players don't seem that interested in disposable characters. Rather personalized epics focused on their characters and how cool they are.
Exploration. The TSR editions of D&D had a lot of focus on exploration which WotC trashed. In 5E exploration is a joke. What is there is toothless to the point of wasting page count to related the "rules" to the player and DM. There are so many skip buttons from class and subclass abilities and spells that obviate what minimal challenge there is in exploration in 5E that you either need to house rule the hell out of the relevant systems to make it worth engaging with or you just skip it entirely.
Variety of game play. Dungeons, wilderness exploration, town and city politics, domain management, regional powers, eventually gaining great power and potentially becoming a god. I liked that the default assumption of what you'd be doing in the game changed as you leveled. This is mostly from the Basic line but there was some support for some of these in AD&D. That variety has been collapsed down to almost nothing. The only thing that changes as you level is the names of the monsters you kill.
Unexpected and weird. Though I didn't play 2E, I absolutely love the settings. Al-Qadim, Dark Sun, and Spelljammer are three of my top four. Mystara / Known World is another. I love the weird and unexpected of the OSR. Not the childish edgy for edgy's sake stuff. But the honestly bizarre. I really miss that about old-school D&D. It simply doesn't exist in 5E. DM's can convert stuff and there's some really good 3PP stuff that scratches a similar itch, but the default 5E is bland as bland can be.