This is a big question (and apologies in advance for a big answer that is still insufficient in so many ways)
For pelagic marine species, it can be very difficult to confidently match specific sounds to a specific species. Not only must you confirm the presence of a species in the area at the same time of a recording, but one should also exclude the presence of other species. Localization of sounds to the same location as the sighted animals as well as sufficient visual observation from a high platform (to ensure good visual coverage) strengthen the confirmation of single-species identity in a recording (see discussion in this paper suggesting an early recording of northern right whale dolphins may have wrongly attributed whistles to this species). In addition to knowing that a sound with certain characteristics can be attributed to a given species, you also need some degree of confidence that the sounds are not made by other species (see this example of sounds produced by sei whales that are extremely similar to sounds produced by fin whales).
For most species, initial classification efforts begin with pairing a sighting of a single-species group with a near-field recording. The lower the frequency of the call of interest, the greater the range of detection and the more complicated it can be to confidently attribute sounds to that species.
For delphinids, vocalizations can more readily be attributed to a given species (existence of species in captivity, smaller transmission range and greater ease of localization). The complication for delphinids is that they have large variability in their vocal repertoire combined with an overlap in call characteristics between many species. For some delphinid species, some of their calls may be particularly unique to that species (see Soldevilla et al. 2008). In her description of these species, Soldevilla refers to original work describing the echolocation clicks of these species, and her work included recordings of animals sighted in proximity to hydrophones. Once calls can be confidently identified to a particular species, and clearly differentiated from other species, researchers can then rely on these call characteristics to examine larger archives of recordings from autonomous hydrophones. To date these species-specific call characteristics have focused primarily on echolocation clicks, but there are some cases of dolphins with more stereotypic characteristics (such as rough-toothed dolphins).
For species where we are not (yet) able to identify species-specific call characteristics, researchers have relied on machine learning tools using a larger sample of recordings (see examples by Frasier 2017, Rankin et al. 2016, Roch et al. 2008). Again, classification techniques should be inclusive of all species in the region (and up for consideration).
For baleen whales, their lower frequency sounds travel greater distances and localization is more complicated. Due to this, most research has focused on their stereotyped ‘song’ vocalizations. Many of the initial descriptions were made from handheld hydrophones in the presence of a given species (such as blue whales in Chile by Cummings and Thompson, 1971), but then these have been confirmed time and again over the years. Localization of some baleen whales using tag data (example given by OP), sonobuoys (see McDonald et al 2005 or Rankin et al. 2005 as examples) or even towed arrays (for minke whales). This gets much more complicated when addressing non-song vocalizations (such as the similar fin/sei whale call noted earlier in this answer), but classification of non-song calls may be possible using novel analytical approaches, such as using Wigner-Ville tranforms to discriminate downswept calls produced by blue, fin, and se whales(see Ou et al. 2015).
The OP notes that publications ‘scant detail about each call type described’. I think it is good form for any paper relying on classification of any acoustic detection as the foundation of their work to cite a solid reference that links that call to that species. That could range from a paper that unequivocally matches a call to a species (such as with minke boings) or cites a specific classification technique (ideally along with the confidence estimates for that technique). Where there remains any question in the source of a call, it would be wise to use caution in how this is approached in publications (a good example is how Baumann-Pickering et al 2013 address beaked whale echolocation clicks that are not yet confidently identified to a specific species).
Besides doing this in our own publications, we can encourage this through our peer-review.