Unfortunately, yes. For example, there are many areas/communities here in the States wherein, regardless of how clean or well-dressed I may come across, I'll still be held suspiciously under preconceived notions relegated to members of "the Other", while a member of the in-group, on the other hand, will not be subjugated under those notions. However, this is a naturally occurring phenomenon born from societal interaction between different social groups. It's fallacious to expect members of a majority to self-regulate in their own country without taking into account their socio-economic stability.
"Privilege", in this sense, therefore, is to be relatively expected. Instead, the "privileges" that should be concentrated upon, and are rather worthy to even articulate, are those that have the capacity and the ability to greatly damage another social group. For example, the taking of diverse and non-Western philosophies and regurgitating them under different perspectives without giving them proper credit. Not only is that theft, but it's also a "privilege" born from operating under the global perspective that such non-Western "trinkets" are free-game. Examples like these are worth addressing.
However, White girls in saaris, for example, which goes back to the "#stopwhitepeople2015" crap, are the least of my worries. The concern is this, or at least my concern as a conscious Hindu, address the realities that have the ability to affect a group collectively and greatly damage their socio-economic and socio-cultural standing in the long-run. White girls in saaris, however, is not as damaging as missionary activity in the Subcontinent whose conduction by mostly Whites acts as a measure through which the "downtrodden" are seen as "finally worthy" to be "saved". And what many White Hindu sympathizers also forget, is that many members of such camps deem non-Brown converts to Hinduism as misappropriationists. So while they may be "fighting the good fight", there is the obvious reality of facing backlash from your own in-group members (i.e., White person telling a "born hindu" not to wear hats inside the temple, and others misconstruing it as racism; and many White Hindus arbitrarily conceding that such an act is, indeed, racist and the White Hindu should only relegate such rules to other White attendees).