Which characteristically Baroque elements of music composition still have currency in the 21st century? Are there any 21st-century composers or songwriters who still compose primarily or largely in the Baroque style? What makes their work especially “Baroque”?
The short answer is yes (and I have occasionally written such music myself). Musical styles never really die, they just fall out of general fashion.
It should be noted that there are a couple categories of music that might be considered in an answer.
First off, and perhaps least authentic, are what might be termed "fusion" styles -- mixtures of baroque (or classical) with modern elements. These are usually called Neobaroque, or Neoclassical, and the example I'm thinking of is Stravinsky's Pulcinella, which is said to be neoclassical. Another example might be Joaquin Rodrigo's Fantasía para un gentilhombre which is based on Baroque guitar exercises. I have a feeling you're looking for something more authentic than these, but its worth remembering that all attempts at authenticity will be colored to some extent by the intervening centuries of music history.
Next off is people who who will write an occasional piece in a more authentic Baroque style for some specific reason or affect. This could be for pedagogical reasons, such as students studying Bach chorale harmonizations, or for its affect, such as a film score for a historical film, trying to evoke a particular historical mood. One piece that springs to mind in this category is Karl Jenkin's Palladio (also known as "Diamond Music" due to its past prominence in TV ads for a certain jewelery company) which does a wonderful job of evoking the feeling of Vivaldi. Another example that probably falls closer to the pedagogical category is the Lady Gaga Fugue.
Finally, what I think you're after is composers who regularly (though perhaps not exclusively) write in a relatively authentic Baroque style merely because they like this style of music. I know that such composers exist,
though I can't remember any of them at the moment. They tend to be relatively obscure, and my impression is that the larger composer community generally ignores them, because they are always looking for newer techniques and styles. An example of one such composer whose name I have come across before is Michael J. Starke.
Another fact worth considering is the economic realities of supply and demand. Patronage for a supply of newly-composed, authentic Baroque-style music is probably pretty sparse, so most professional composers don't really have a good regular outlet for this, even if they wanted to. Thus most people who would qualify as answers are just as likely to be composers "on-the-side" who dabble in it as a personal hobby, rather than professional full-time composers.
Edit: A more general name for the phenomenon of writing in a past musical style is Musical Historicism. Wikipedia mentions, as an example:
Roman Turovsky-Savchuk, whose original lute and viola da gamba compositions in the baroque style were sufficiently convincing to be mistaken for works by masters of the composer's own mythopoeic invention (Colburn 2007), and led to accusations of "trivializing musicology" (Smith 2002).
With the resurgence of historically-informed performance, it is not surprising that there is also a resurgence in historically-informed composition. Indeed, the above Wikipedia page also mentions an international society called Vox Sæculorum.
Vox Sæculorum is an international society of contemporary composers writing in the Baroque style established in 2006.
1). Whenever a composer makes use of counterpoint within the context of a tonal chord progression, or writes a fugue, the composer is more or less going straight back to the compositional technique of the Baroque composers such as J. S. Bach, G. F. Handel, or Telemann. So there are quite a few examples of these compositional techniques in 20th and 21st century music, even if they are done with contemporary musical instruments or if they are using a harmonic vocabulary (for instance, using jazz chords) that is more modern than that used in the Baroque era.
2). There is a lot of contemporary movie soundtrack and video game soundtrack music that consciously emulates the Baroque style and Baroque instruments. You can find this in particular in genres like fantasy, sword-and-sorcery, and any movies or video games set in an "ancient" setting. In movies and video games that you would consider to be in a medieval time period, the new music composed for the soundtrack is often in the Baroque style, or perhaps a Renaissance style.
3). Metal. These days, occasionally you find elements of Baroque-style composing, usually performed on electronic instruments and orchestral digital sample libraries, in heavy metal music.
4). Today there are Baroque chamber orchestras all over Europe and the New World. They perform music of the Baroque composers on replicas of the instruments used in the Baroque era, in historically-informed performance style. On occasion these groups will commission a contemporary composer to write a new piece for them to perform. Such pieces might be strictly in the Baroque style, or they may be in a much more contemporary style but written to be performed on Baroque-period instruments in historically-informed performance.
5). One composer whose name comes up frequently as using Baroque-period instruments in new works, hearkening back to the Baroque style, is the recently-deceased British composer John Tavener (1944-2013).
There's also P.D.Q. Bach, who is J.S. Bach's last and forgotten son, but was way ahead of his time, many of his pieces clearly informed by 20th/21th century music. My impression is that compositions by P.D.Q. Bach, no matter their many failings, enjoy broader notoriety than, for example, Professor Schickele's pieces, who's an award winning 21th century baroque composer. The race is on, great to see some healthy competition in the baroque music space!