Why is Shabbat the only day of the week that is given a name in Tanach where as all the other days of the week aren't given a name in Tanach?
As @DoubleAA mentioned, "Shabbat is not a day of the week. It's a holiday which happens to always fall on the same day of the week. The day of the week is Yom HaShevi'i".
There are many verses that prove this point. But, perhaps the clearest example comes from Vayikra 23:2-3. I will summarize the concept rather than do a literal translation.
Verse 2 starts with an introduction saying "These are my holidays that you should make holy". In other words, the Torah is already stating that what will be coming is a list of holidays.
Verse 3 says - You shall work for 6 days, and the 7th day shall be a Shabbat. Meaning, a "rest day". Note that the day itself is referred to as just "the seventh day". It is still a holiday, that doesn't change things. The holiday itself is called "Shabbat", and it occurs on the 7th day of the week.
See also Sforno's explanation of the phrase וביום השביעי שבת - he says it means that you should rest from your work. It does not mean, "The seventh day should be called 'Shabbat'."
Assuming permission from @Chaim, I'm copying the verse that he cited in his comment:
וַיְכַל אֱלֹקים בַּיּוֹם הַשְּׁבִיעִי מְלַאכְתּוֹ אֲשֶׁר עָשָׂה וַיִּשְׁבֹּת בַּיּוֹם הַשְּׁבִיעִי מִכָּל מְלַאכְתּוֹ אֲשֶׁר עָשָׂה: (בראשית ב:ב)
My translation of above verse from Breishit (Gen. 2:2)
G-d finished, on the seventh day, all the work that he did. He rested on the seventh day from all the work that he did.
Generally, when we learn Torah, and we want to understand the definition of a word, we look for its first occurrence in the Torah.
There are two points to be learned from the above verse. First, you can see that the 7th day of the week is referred as exactly that - the 7th day. You don't see it called "Shabbat" anywhere.
Second, we see the word וַיִּשְׁבֹּת which has the root form שבת meaning "to rest".
I once heard in a Shiur that by specifically not giving names to other days of the week we are given easy opportunities to merit in the Mitzva of "remembering the Shabbos" on a regular basis. By saying yom rishon, yom sheni etc. it is automatically referencing rishon and sheni to what - to The Shabbos.
[DanF adds:] This concept is an integral part of the 4th (of the 10) commandment, זכור את יום השבת לקדשו - Remember the Shabbat day to sanctify it. Many siddurim include mentioning this verse in the sentence just preceding the recital of the daily Psalm in Shacharit.