In brief, there are specific things to listen for in evaluating a singer's vocal accomplishment. Whether one can hear those things is another matter. It's useful to have sung oneself, or to have worked with singers. Here are the things I look for:
1. Ease of vocal emission. Does the sound seem to spin out freely, or is there a sense of forcing or unnecessary effort?
2. Consistency of vocal emission. Does the sound seem to be produced with similar ease and quality at all times, or do changes in pitch, volume, and velocity cause distortions in these aspects?
3. Evenness of scale. Is there consistency in the timbre and production of the voice throughout its range, or are there distinct differences in quality or abrupt shifts in sound as the voice moves up and down its range?
4. Freedom and consistency of vibrato. Does the vibrato have a quick, even pulsation, fairly consistent despite changes in pitch and volume, or is it irregular, slow, quavery, or excessively wide (obscuring the pitch of the sung note)? Anomalies in the vibrato are due to muscular interference, wear, or fatigue.
5. Control of dynamics (volume). Can the voice move freely among dynamic levels, swelling and diminishing the tone at will without compromising evenness of production? This is a hallmark and test of a fine technique, and a requisite of expressive singing.
6. Flexibility or agility. Can the voice move quickly and freely from note to note? Voices vary naturally in this respect; a lack of agility may or may not indicate poor technique, but it is certainly a disadvantage.
These are the basic elements of vocal technique, all of them quite perceptible. They are objectively present in voices and are not, for the most part, matters of personal taste. The only partial exception is the vibrato; vibratos naturally vary considerably from singer to singer, and a vibrato which strikes a particular listener as unattractively wide or prominent may or may not indicate a technical deficiency. Wide vibratos have never been prized, however, for the reason that they tend to obscure pitch, if for no other.
Beyond technical considerations are musical and stylistic ones - but that's an immense subject.
Bringing this back to Stracciari in particular, he was a great singer in that he was virtually flawless by all the above technical criteria. I would submit that he was also a fine musician with excellent style in the repertoire he sang. If you would care to compare him directly with other baritones, I'll refer you to my post of 3/4/15 under the thread "Which singer best represents each fach?", where I compare nine other baritones of various generations from Battistini to Milnes, using the aria "Eri tu" from Un Ballo in Maschera
as a test piece.
That was an exercise I found enjoyable and enlightening.
(As a postscript, I want to say that I do see the OP as asking for our personal tastes and not necessarily objective proofs of anything. Stracciari is a favorite of mine who I think is as fine a baritone as I've ever heard, but there have been others; among Italian baritones I also love Battistini, a great representative of the bel canto school of the 19th century, and Pasquale Amato, who unfortunately burned out too soon, apparently trying to compete with the powerhouse Titta Ruffo.)