Yes, it can be done unless concentration is required to do so
Bardic music states:
Starting a bardic music effect is a standard action. Some bardic music
abilities require concentration, which means the bard must take a
standard action each round to maintain the ability. Even while using
bardic music that doesn’t require concentration, a bard cannot cast
spells, activate magic items by spell completion (such as scrolls),
spell trigger (such as wands), or command word. Just as for casting a
spell with a verbal component, a deaf bard has a 20% chance to fail
when attempting to use bardic music. If he fails, the attempt still
counts against his daily limit.
As stated in the Dragon Compendium p. 15 (the only official source material for the Dvati in 3.5e):
Language: The dvati language requires two creatures speaking simultaneously to properly use it, and its unique alphabet consists of only eight characters. The character's alignment—upside down, mirrored to the left or right, or turned to the side—determines its exact sound. If a dvati is somehow separated from his twin, he uses Common to communicate with others.
Automatic Languages: Common and Dvati. Bonus Languages: Any. Dvati requires two creatures speaking simultaneously. One twin provides a description of actions, while the other supplies words and descriptions of things. The two weave together to communicate in a uniquely dvati manner. Other creatures can speak this tongue, but two Dvati-speaking creatures must work together. Otherwise, communication via this language is impossible. In most cases, dvati use Common to speak with other races.
So unless you are trying to sing Dvati songs (since "any" language as bonus is mentionned, Dvati characters are not stuck with only Common when they are alone by RAW in my opinion because it was only stated in the Dvati language section and that it was probably referring to automatic languages only, when I played that race back then my DM ruled it was only common so the DM decides how he interprets it and as always check with your DM), each of your Dvati can sing a different song, as for the concentration part you mentioned p. 14 states:
are born identical twins, with the pair
sharing a single soul between them.
Anything that is physical can be done separately but anything that is purely mental and requires concentration (such as some spells and abilities) is not possible separately for balance in my opinion because they share only one mind. Both characters can sing two different songs at the same time if it does not require a concentration check. (except in Dvati language)
However since bardic music isn't strictly limited to singing (why not sing in common, use only an instrument or even whistling?) and that War chanter music follows the same rules as bardic music (a dragonsong lyrist can use song or poetics to invoke the power of dragonsong, so voice is necessary for dragonsong though but concentration is not needed so it works unless your DM limits you to common, but that only applies to dragonsong since it must be done in Draconic if I read correctly) it can be done by other methods:
Bardic music states:
they can all be activated by reciting poetry, chanting, singing lyrical songs, singing melodies, whistling, playing an instrument, or playing an instrument in combination with some spoken performance
- As you yourself shared the link to the ability pool question, p.17 of Dragon compendium states:
A pair of dvati twins shares all class abilities and spells
slots between them. For example, a 3rd-level dvati bard can use
bardic music three times per day in total, not three times for each
- I also don't believe you can stack that many music effects, I only responded to the general question.