It is the major problem of all cognitive sciences to deal with subjectivity. On the other hand, in order to figure out what is cognition or psychology, one must ultimately deal with this problem. As a starter, I wonder if there is any study about identicalness of perceptions among subjects. Are the perceptions of two different subjects on an object identical? In other words, can the perceptions of two subjects about a tomato differ from each other? One may experience the redness of tomato as what the other call blue.
Can We Compare Subjective Experience?
Consider this pain scale, variations of which are commonly used in medical settings:
If two people answer "6 - distressing, miserable pain," can we reliably compare the subjective pain intensities between the two individuals? How would we know that one person's experience of "distressing, miserable pain" is the same intensity as the other's?
Supertasters and Taste Perception
What's helped us to answer a question like this comes from Linda Bartoshuk's work on taste perception in so-called supertasters vs. normal tasters.
Bartoshuk famously uses blue dye to reveal the number of tastebuds on participants' tongues. As you can see below, supertasters have a lot more tastebuds than normal folks (and what Bartoshuk calls "non-tasters"). As a consequence of this, we see variation in taste perception.
What's important is that the number of taste buds provides a reliable physical/genetic basis by which to predict taste perception (e.g., bitterness intensity). So we would expect reports of bitterness intensity like so: supertasters > normal tasters > non-tasters.
Consider this 9-point scale assessing bitterness intensity:
If a supertaster and taster both rated a 5 on this scale, we'd wonder why there wasn't a difference. In fact, how could we know that a 5 for a supertaster is the same intensity as a 5 for a normal taster? Perhaps a 5 is twice as bitter for a supertaster than a normal taster, but this scale is not suitable for determining that.
Without getting into too much detail, the solution to between-group comparison is cross-modality matching. Basically, we ask participants to compare their experiences of bitterness to, for example, the brightness of a light.
Importantly, perceptions of brightness are presumed to be independent of perceptions of taste (i.e., having more/less taste buds isn't related to perceptions of brightness). So variation in brightness between the two groups will be the same on average. Thus, it can serve as a potential means by which to compare subjective intensity of taste perception.
Bartoshuk uses this example with sound that I find easy to understand:
We put earphones on everyone and ask both groups to adjust the loudness of a tone to the sweetness of the soda. If hearing and taste are really independent, then the average perception of loudness should be the same in the two groups, but subjects with the high density of taste buds set the tone to 90 decibels (about the loudness of a train whistle), and those with the low density set the tone to 80 decibels (about the loudness of a dial tone). As we know that a tone of 90 decibels sounds twice as loud as a tone of 80 decibels, we can conclude that the soda tastes twice as sweet to those with the high density of taste buds. We have identified a systematic difference in taste intensity between these two groups.
You can also use imagined sensations. For example, Fast (2004) has found that:
supertasters matched the bitterness of dark chocolate to the brightness of low beam headlights at night, while nontasters matched it to the brightness of the moon.
Consider this scale, which uses cross-modality matching to assess perception of candy sweetness intensity between supertasters (ST) and non-tasters (NT):
When using these matched modalities, you get systematic differences in sweetness perception that you otherwise wouldn't see using the standard 9-point sensory scale above.
I didn't give a particularly comprehensive outline of cross-modality matching, so for more information, consult the sources below.
Bartoshuk, L. (2014). The measurement of pleasure and pain. Perspectives on Psychological Science, 9(1), 91-93.
Bartoshuk, L. M., Fast, K., & Snyder, D. J. (2005). Differences in our sensory worlds: Invalid comparisons With labeled scales. Current Directions in Psychological Science, 14(3), 122-125.
Kalva, J. J., Sims, C. A., Puentes, L. A., Snyder, D. J., & Bartoshuk, L. M. (2014). Comparison of the Hedonic General Labeled Magnitude Scale with the Hedonic 9-Point Scale. Journal of Food Science, 79(2), S238-S245.
These are some of the common ways that psychologists use to test subjective experience:
Introspection. Introspection has fallen a bit out of fashion after the hype surrounding mathematical psychology, but in some areas introspection still is a valid method, especially if other approaches are impossible or unethical.
A famous psychological self-experiment whose result rely solely on introspection is that of Niels Bierbaumer, who let himself be injected with a deadly dosis of curare, which he survived because he was mechanically ventilated by a respiration therapist.
Interviews. Just ask what a person feels or thinks. Quite often they simply tell you the truth.
Questionnaires. Some are straightforward, some try (some even succeed at) hiding what information they are after. Compared to interviews they have the advantage of allowing only standardized (questions and) answers, which can then by evaluated mathematically.
Implicit tests. An example is the implicit association test, in which different reaction times to different stimuli (e.g. words or images) are interpreted as a person's bias towards those stimuli.
Taking your example, you might present a person with the word "Tomato", an image of a tomato, or the actual object, and measure how quickly they click on the word "tasty", and compare that to how quickly they click on the same word when they are presented with, for example, a banana.
Observation of behavior. Either in an experimental setting or a field study. I don't know what the correct term is for observing normally occurring everyday behavior (in contrast to artificial behavior like clicking buttons when you see words on a screen), but I will give an example:
You invite people for a study that you say is about emotions in movies. That study is only a "cover story" to the real test you are doing. Participants have to watch a movie and (that is the fake study) fill in a questionnaire about it. What people don't know is that some of them get to see a happy movie, others see a sad movie. The movie is long, and the questionnaire is even longer (and you set the experiment to fall into people's lunch time, so they arrive hungry), so you give them something to eat between the movie and the "test": there is a bowl of tomato salad, a fruit salad, some bread, and so on. What you are interested in and what you actually observe and measure is, wether people are more inclined to eat tomato salad after a happy or sad movie. (The number of participants and their random distribution to the two movies will average out the fact that certain individuals like or dislike tomatoes.)
There are many other ways to indirectly study human (and animal) emotion and thought, and it is part of the ingenuity and creativity of scholars to invent ever new study designs. There are too many of them for me to list them here (and I certainly don't know half of them), but if you are interested, there are many books in every scientific library that teach about designing studies, and many articles (some freely available online) that show how research is actually done.