What animals?
I'll start with a broad unsupported hunch: probably almost any animal? The details and success are however likely to be a matter of which features (not easy to know beforehand in an understudied species) you use and the stats/ML methodology. When the OP talks about identification, I'm hereon assuming they mean automated identification.
For instance this review by Carlson, Kelly & Couzin 2020 points out various animals that show individual call production (based on evidence that others can identify them only through sound). In their paper you see the literature has evidence of individual recognition in birds, and much smaller mammals (like marmots!). I know there's some evidence for bats having unique echolocation calls too in Yovel et al. 2009 (I say some evidence because N=5 bats if memory serves right).
Hunch: In cases of failure to detect individual signatures, it may be genuine lack of variation or IMHO our genuine lack of understanding on the features to use (esp. for short sounds like clicks or sweeps of a few micros to milli-seconds duration). Here too we might expect to see a vertebrate/mammalian bias. A lot of the voice recognition work is inspired by and based on work in human biometrics.
To what extent?
The question about the stats/ML methods: are the studies using a 'closed-pool' (the 'easier' case) or an 'open pool' (the 'tougher' case) to test the effectiveness of recognizability? A closed pool means there were 10 individuals that form the sample and recognition is always checked by presenting the algorithm with one of the 10 already 'seen' individuals. An open-pool is when 10 individuals were used for training, but the presented individuals can also be 1 previously 'unseen' individual. Here I remember reading that methods used for human voice recognition (based on Gaussian Mixed Models and others) typically show a quality-quantity tradeoff (Encyc. Of Cryptography and Security) . The methods can tell individuals apart in small groups easily but struggle with larger groups. Though a vocalisation may seem stable, it is important to consider ontological change (as animals age the voice change) as well as short term changes (blocked nose, etc).
My suspicion is that many animal studies often have (understandably, just given the insane logistics of training and working w animals!) low sample sizes (few individuals tested), allowing stats/ML techniques to show 'good performance'.
The cues that the stats/ML tools pick up may not always correspond with what the animals are using. Recording animals consistently and 'equally' even in lab settings isn't easy (let's not talk about field recordings now). For instance, a consistently 'loud' animal can be easily identified by eye or algorithm, but is this because of a unique favourite perch position close to the mic? If the identification is for census purposes this is a moot point perhaps, but it may have unintended consequences:). The classifier may sometimes tune in on other consistent cues like stream noise, other adjacent animal calls instead of the actual target species' call! (see Stowell et al. 2018 for a demo on pipelines and things to be wary of while training a classifier).
Note: the OP's question was admittedly asking for other species aside from the one mentioned in the question. The answer here intentionally deviates from creating a list as it doesn't add too much in terms of general know-how on this SE. Also refer to this Meta discussion and participate!
