In general, what you are referring to is what we call numeraire pricing. This means that the prices, as you mentioned, are relative to one of the goods (you can pick any good as a numeraire, which is how we measure prices through the whole economy).
The reason for this is Walras' Law states (basically) that since we have a bunch (say $N$) goods in the economy, and say we have a bunch (again, $N$) markets in the economy, then solving all of these will leave one redundant equation. This applies in particular to general equilibrium because there are essentially more equations than variables, and hence you have one (or more) redundant variable(s) at the end.
To get a little more technical if you want: if money is something that we can consume, then we would insert cash as an additional good and make that our numeraire. Thus we can have "money," in the sense that it has a value as a bubble, and we can include it as a good in our utility.
To answer your questions about Veblen/Giffen effects, we can still observe these effects, they have nothing to do with money. These effects in their essence are talking about goods that we buy more of when the price increases. Even if this price is in terms of "money" or a numeraire, the point is that we buy it when it's more expensive to us, so we push other things that we might need out of the way in order to buy this.
A quick example (don't think too deeply about it, it's just an illustration) would be college in America, using, say, $1 McDonald's McChicken sandwiches as a numeraire. An argument could be made that each year more students attend college yet the price in terms of McChickens goes up each year ((i.e., consumption increases, as the price goes up). This could be described by the Veblen effect, where we maybe don't want to be perceived as someone with "only" an associate's degree or high school degree (not that those are bad, but it's just the way things are in America). If I measured in dollars or McChickens, the Giffen or Veblen effect doesn't change.