Before we talk about "in vain" we need to talk about "take". The passage is:
לֹא תִשָּׂא אֶת-שֵׁם-יְהוָה אֱלֹהֶיךָ, לַשָּׁוְא:
The first verb, תִשָּׂא, is literally "lift up", the same word as in, e.g. Psalm 121, "I will lift up mine eyes". This isn't "take" as in "acquire", probably; it seems like a different word would have been used (lakach or kanah). We seem to be talking here about "holding God up", so to speak.
For what? שָּׁוְא, translated here as "vain", is also translated as "emptiness". This is consistent with not making light of God's name, such as in the near-ubiquitous "OMG". It's also consistent with "fake oaths", like when someone says in anger "I swear to G... that I'll (do something awful that he won't really do when he calms down)". Either way, you're involving God in a joke or a false utterance, not treating God's name with seriousness.
Rashi, informed by the targum (Onkelos), understands the whole phrase as meaning not to swear false oaths. He writes:
You shall not take the name of the Lord, your God, in vain: You shall not swear in vain by the name of the Lord, your God. — [Onkelos] לַֹשָוְא -[This word appears twice in this verse.] (The second [mention of לַֹשָוְא is an expression of falsehood, as the Targum [Onkelos] renders: לְֹשִיקְרָא, as it says [in Shavuos 21a]: "What constitutes a vain oath? If one swears contrary to what is known, [for example, saying] about a stone pillar that it is [made of] gold. (The first [mention of לַֹשָוְא is an expression of vanity, as the Targum [Onkelos] renders: [לְמַגָּנָא].) This [refers to] one who swears for no reason and in vain, [for example making an oath] concerning [a pillar] of wood, [saying] that it is wood, and concerning [a pillar] of stone, [saying] that it is stone. — [from Shevuoth 29a, Mechilta]
An interpretation of swearing of oaths is consistent with "lifting up" God's name; when taking an oath one invokes God, essentially as a witness. This is not a light matter; if you speak falsely, you are involving God in that falsehood too. To do so for a serious transgression is bad enough; to do so for a trivial matter seems even worse -- not only are you bringing God into a falsehood, but you're bringing God into pointless, trivial falsehood.
Whether you consider identifying oneself falsely as being one of God's people would seem to depend on what exactly you said. "Do not claim to be an Israelite if you're not" seems like an unlikely interpretation to me, though claiming explicitly to be one of God's people, using the divine name, would be consistent with the text.
Conclusion: the text supports the following possibilities:
Either way, it is about the (or a) divine name, and probably not a more-general tribal, social, or other identification.
Please note: This answer was written for a neutral, academic audience
and is not intended to be interpreted in the context of a religious
belief or doctrine.