I've noticed that what some languages refer to as "soft k" and others as "soft t" seems to be the same sound. Is it so?
I was able to find a wiki page discussing the "soft g"/"soft d", but nothing on this case.
IPA is a fairly limited tool to describe this area of pronunciation. Like /a/ for open vowels or /r/ for rhotic consonants, the IPA symbol /c/ can be used in a broad as well as a narrow sense.
One issue is that none of “/tʲ/“, “palatalized t”, ”/kʲ/“, palatalized k”, “soft k", “soft t” refers unambiguously to a single phone (a sound defined in terms of its phonetic or articulatory properties, rather than in terms of its function in a language’s sound system). The symbols /tʲ/ and /kʲ/ are in slashes, meaning they denote phonemes, and phonemes are language-specific. Terminology like "soft" is even more vague: you could say from a certain point of view that the English word “patience” has “soft t” and “soft c”. The term "palatalized" has multiple senses: a phonetic sense, a phonological sense, and a historic sense.
It is possible for a language to have a contrast between what can be described as a palatalized t phoneme and a palatalized k phoneme, in which case we can transcribe these phonemes as /tʲ/ and /kʲ/ respectively.
Irish is a good example, although the palatal(ized) counterpart to /k/ in Irish is often transcribed as /c/ rather than as /kʲ/. (user6726 posted a comment beneath my answer to another related question, What are the differences between palatal consonant and palatalized consonant?, suggesting that "under classical assumptions about binary features [[kʲ] and [c] (likewise [ç] and [xʲ]] could not contrast". Janus Bahs Jacquet posted a comment below saying "Irish /kʲ/ is actually [kʲ] or [k̟] in most northern dialects, that is, it is decidedly still (pre-)velar. Conversely, it’s fully palatal [c] in southern dialects").
Russian is another example, although a weaker one since in native vocabulary, Russian palatalized and unpalatalized velars are generally in complementary distribution.1
For these languages, we can justify the use of the notation /tʲ/ and /kʲ/ by noting that Irish and Russian seem to in general organize their phoneme inventories in terms of pairs of palatalized and velar(ized) consonants. Thus, on the level of their phonemic system, it is reasonable to say that there are /tʲ/ and /kʲ/ sounds just as there are other palatalized sounds like /pʲ/, /mʲ/, /lʲ/. On the phonetic level, things may be more complicated.
In theory, the sign [ʲ] in IPA phonetic notation is supposed to describe a secondary articulation of palatalization, so [tʲ] and [kʲ] should respectively represent a palatalized coronal plosive and a palatalized velar plosive.
Phonetically, the issue with the IPA palatalization diacritic in regard to [tʲ] and [kʲ] is that it's not actually so common for palatalized versions of these sounds to be realized with a "secondary" articulation present alongside a clearly distinct "primary" articulation (as we see with palatalized labial sounds like [pʲ]). Instead, since coronal, palatal and velar sounds are all articulated with somewhat similar areas of the tongue, "palatalization" of [t] or [k] tends to result in the sound being realized as a fully palatal stop, as a stop with some kind of intermediate place of articulation, or as some kind of affricate.
The IPA letter [c], when used to represent a phone rather than a phoneme, is vague: it is defined as a voiceless palatal plosive, but it is not clearly defined whether it is closer in quality to a [tʲ]-like sound or a [kʲ]-like sound. The transcriptions [c̟] or [t̠ʲ], mentioned in the Wikipedia article Voiceless palatal plosive, are ways to be more specific.
But many phonetic transcriptions aren't that specific (phonetic transcriptions can be "narrow" or "broad", and in broad transcription details like this can be left out), and in phonemic transcriptions, it can even be considered bad style to use diacritics that are not needed to mark a phonemic distinction in the language. So in practice, the IPA letter is underspecified in terms of what phones it represents, meaning might represent different sounds in different languages.
The phonetic value of a sound transcribed as /c/, /tʲ/ or /kʲ/ in a language that does not have a contrast between these symbols could be various things. A sound transcribed as /c/ in one language could conceivably be the same as a sound transcribed as /tʲ/ in a second language, or /kʲ/ in a third language.
In the case of Russian /tʲ/, Wikipedia says "Soft /tʲ, dʲ, nʲ/ are laminal alveolar [t̻ʲsʲ, d̻ʲzʲ, n̻ʲ]. In the case of the first two, the tongue is raised just enough to produce slight frication as indicated in the transcription", citing Jones & Ward (1969:104–105 and 162) (Russian phonology - Wikipedia). I don't know what more recent descriptions say about Russian.
As for Latvian "ķ", I haven't found a detailed discussion of its pronunciation but it seems to be generally described as a true palatal [c].