Why is Batman not so careful about his secret identity in the Justice League?

First of all, in our current society, secret identities make no sense at all.

• In the 90 batman animated story, Talia-Al-Ghul was able to find Batman's secret identity through detective work.
• In Man of Steel, Lois found Kal-El even before knowing Clark.
• In B vs S, Lex Luthor found Batman and Superman secret identities, he also found information about the rest of the current Justice League.
• In Justice League, Bruce Wayne was able to track all the remaining Justice league members.
• In Civil War, Stark was able to track Peter Parker.
• In real life, Facebook was able (without even trying) to find an escort girl secret identity.

In a society with the Internet and heavy social network, it is not possible anymore to keep a secret identity. More if you are a public figure. That's the main challenge movies universe have now to acknowledge. Especially when talking about heroes that relies heavily on a secret identity.

Now about Batman

Well, my main impression is that he doesn't care anymore.

Twice in the movie, he was ready to let himself killed 'for the team'.

The plane scene clearly shows how tired he is. And earlier we learn that he's been the Bat for 20 years... (It was even my impression that at the start of B Vs S, he already had retired, and came out of retirement due to Man of Steel events (and stays due to Flash warning). He might even get sloppy.
You can add the fact that he knows that Lex Luthor already found out. And may have already planted 'information bombs' that can go off at any time. His secret is on borrowed time anyway.

But why the mask if he doesn't care?

Part of it is to keep his personae. He is the Bat. It's the bat that induces fear, not the man.

Most superheroes use a secret identity to protect loved one. Bruce chooses the mask to induce fear. For what we know, he doesn't have loved one to protect (except Alfred, and neither of them have dates, as stated in the movie), but he still needs to induce the fear.

Another part is that the mask gives protection.

Why do you think bikers wear a full helmet? They don't care about any secret identity.

Basically the DCEU movies don't concern themselves much with the topic of secret identities, hence the neglect of that topic by the characters.

Batman vs Superman would have been the perfect opportunity to cover this topic and how secret identities hold up in the modern world, but they never really touched the topic - Luthor simply identified both Superman and Batman's other identities, but we are never told how. Similarly it's basically taken as a given that the audience already knows about their dual identity.

In-universe Batman may indeed not care anymore (which would mean he also knowingly endangers Alfred, which would be somewhat out of character), but what about the others? Even for WonderWoman who is concerned about the picture Luthor has, it's unclear why she wants it back. If it is for anonymity,the topic is brushed over. Basically, I'd attribute this to writer laziness - they are not interested in the topic in general, so it's not of much concern when they write scenes.

I also don't buy that it wouldn't be possible to keep a secret identity in our times. Compared to earlier times, the capabilities to uncover secret identities might well be compensated by increased capabilities of (tech-savy) Superheroes. While it was always borderline ridiculous that Superman could keep his identity as Clark Kent secret (and so is in the DCEU), for Batman it shouldn't be a bigger problem than it was back in the day. Sure, there are more CCTV cameras and the occasional snapshot from a bystander's phone is likely. However, biometric face recognition can be hindered by the mask and contact lenses, fingerprints have always been a problem (but can be avoided) and for the occasional picture that could identify him he surely knows how to monitor the web, influence public opinion and hack the source devices to alter/destroy the pictures. The only real threat would be government level organizations if they were to focus on him. And the same could be said about his older incarnations - enough agents following him whenever he is seen would also be a threat to identify where he goes and ultimately who he is.