The two most popular kinds of sidekick are followers and pets. I want to use general terms here because old schoolers will get hung up on the distinctions between hirelings, henchmen, and retainers while new schoolers will get hung up on the distinctions between animal companions, mounts, familiars, and other summons. But that's missing the point.
Obviously, both old school D&D and new school D&D can and do make use of both followers and pets. But they definitely each have a preference. Modern DMs have to choose to add followers into the game, often because they specifically want to add a pinch of old school! And old-minded DMs rarely are prepared for when their younger players inevitably ask to have a pet, and at best might homebrew some "animal taming" procedure to feel better about it. I'm just here to point out that each one is better adapted to the norms and expectations of each play culture, yet are fundamentally variants of the same basic thing.
Followers are better suited to old school play because they're good for carrying items and holding light sources. Modern D&D doesn't care about either of those things. They're also a great backup character if your PC dies. But as far as modern D&D is concerned, an unplanned PC death is basically a complete fail state. It's like the worst thing that could possibly happen in the game and it means that someone, probably the DM, supremely and unforgivably fucked up. Followers are also an active agent to some degree, with their own motive, voice, and concerns about what they're sent to do and how they're treated. This holds potential for interesting social conflict, but it's a type of conflict that modern D&D doesn't have much interest in exploring or validating. Whereas an old school adventurer has to make a choice about whether they're abusive or fair to their followers, modern D&D would simply rather not allow them the opportunity to be abusive to anyone at all.
Pets are better suited to new school play because they don't have motives or agency or much of a voice. They can instead act as a fun accessory for their PC, making them look cooler. Remember, the PCs are much more the focal point of the game nowadays. Time and attention are finite resources, so anytime an NPC is getting the spotlight, it comes at the direct expense of the PCs. Followers are also more mechanically complicated. Pets are simple to run, which is good because PCs are now more complicated to run than in olden times. A pet can just be an extension of the PC, not unlike a mage hand. Of course, they can have personality. The players who want pets the most would all agree that the best part about a pet is when they're cute and fun and charming. There aren't many folks interested in a pet strictly for its practical benefits. That said, they do still have those!
The main utility a pet has in modern D&D is to serve as scouts and spies. That's a type of challenge that remains relevant in modern play, and my own group has to send out a pet to do some reconnaissance almost every single session. The second most common practical use for pets is to have them harass your opponents in battle so you can get advantage on your attack roll. This is a bit cheesy, but the prevailing ruling among the 5E community is to allow this idea (admittedly, probably mostly just so you can placate the player who really really likes their pet and wants it to be involved). Pet as an easy source of advantage means you don't have to be as clever about tactics. No need to work together with your fellow PCs (individual initiative is a modern design choice that already makes that less viable), no need to leverage the environment, no need to really read your opponent for their weaknesses. Just press the "activate pet advantage" button and move on to your attack roll.
I know this all makes me sound curmudgeonly and dramatic, but I sincerely like both of these playstyles and see the value in how each of these sidekick types complement them respectively. Followers make sense if the focus of the game is on navigating a landscape of complex, interactive challenge elements. Pets make sense if the focus of the game is to serve as a terrarium for your blorbos. But of course, everyone who's ever adopted the goblin NPC as a mascot for their party knows that followers can still satisfy blorbo appeal, and everyone who's sought out to tame a unicorn so you can have a powerful mount knows that pets can be treated as a very gameable asset even a rugged adventurer could see the use in.