The term "phoneme" is used in a lot of ways. Most often, people use it to refer to the distribution of "phones" in a given language, where a "phone" is a set of sounds that can be (reliably) auditorily discerned and transcribed using IPA letters. This allows you to maintain a distinction between oral vowel and nasal vowel, or unaspirated and aspirated voiceless stop, but does not encompass "left-side nasalized" vs. "right-side nasalized", or "somewhat aspirated", "aspirated" and "massively aspirated", essentially because this is beyond the scope of what the IPA can handle. These "phones" are what is most closely related to absolute / universal perceptual similarity in sound. One would then find that there are both aspirated and unaspirated voiceless stop phones in English, Korean, Thai, Navaho and Apache. At level of what a trained listener can hear, even though there are only two IPA phone categories available to us (aspirated and unaspirated), it is clear that the aspirated phones of Korean are very different from the aspirated phones of English and from the aspirated phones of Navaho; the unaspirated phones of Thai are likewise perceptually distinct from the unaspirated phones of Navaho. In other words, narrow symbolic transcription of sound properties via IPA yields an intermediate level of granularity, which doesn't even encompass what is audible without using special technology. If you use special technology (e.g. acoustic measurements), there is no limit to the number of distinctions that can be made – acoustic measurements are continuous, whereas categorial judgments are (by definition) discrete.
People will use the term "phoneme" to refer to categorial analysis of sounds as "phones", namely discrete perceived categories of limited granularity. A different (and historically terminologically correct) view is that "phonemes" are a set of phones that potentially "contrast" in a language. On those grounds, aspirated and unaspirated voiceless stops of English are not distinct phonemes (ignoring problematic pairs like capitalistic and militaristic). Under either understanding of "phoneme", you can still compare across languages and you can say that Korean and Hindi are similar in both having a voiceless aspirated alveolar stop phoneme [tʰ], even though there will still be some measurable differences in the acoustic properties of these sounds. Unfortunately, there isn't much data available allowing us to compare acoustic similarity of nominally identical phones across languages. (My standard for a "much" comparison is the amount of non-quantitative transcriptional information available on the languages of the world).
There is a special problem in talking about the similarity of phonemes in the contrastive sense, which is that the properties of an abstract contrastive phoneme abstract away from properties that are not universal amongst the allophones. So in English, the voiceless bilabial stop /p/ is realized with aspirated phones and unaspirated phones -- but what are the properties of the phoneme? Is it aspirated, unaspirated, or not specified? There are many ideologies surrounding this question, for example there is an economy ideology that holds that underlying forms should minimize the number of properties available (which forces an analysis where /p/ is unspecified for aspiration). A competing ideology is based on rule-economy, holding that whatever system yields the simplest rules is the correct analysis, and this forces the analysis that /p/ is underlying unaspirated because stating the distribution of aspiration is formally simpler than stating the distribution of non-aspiration.
To answer your question, then, you would have to make an ideological decision about the properties of phonemes – do they constitute a special class of "unspecified" elements, or is their substance the same as that of phones? If you assume the latter approach, there is a fairly simple basis for comparing sounds in languages. In English, we have the phone(me)s s,θ; in Serbo-Croatian, French, Chinese and Makua there is s and no θ. These languages have sounds that are similar to θ, for example they have both s and t; they also have p which is somewhat similar, but less similar. By simply comparing the categorial properties of these phonemes, you could then arrive at a set of sounds that are "equally similar" to English /θ/. Makua is special in these languages in also having phonemes /t̪, t̪ʰ/, which is more similar to /θ/ than /t/ is (thus for Makua, the sounds that are "most similar" to English /θ/ would be /s/ and one of /t̪, t̪ʰ/).
So in fact, there is a computable answer to your question, provided you make the requisite assumptions about the standard of comparison, and provided that you get reliable information about the properties of the sounds in languages that you want to compare. There is an alternative approach of cross-linguistic experimental perceptual comparison, where you present speakers of language A with a sound X of language B and ask (indirectly) what sound X is most like. This would involve trillions of trillions of comparisons, and hasn't been done.
