Are financial markets "unique" for each "currency pair", or are they simply "translated"?

Economic analysis always requires making some assumptions at some point. The assumptions that you make should try to fit reality the best that they can.

Regarding your specific situation, there is a term in economics that might be helpful in describing what's going on: "Market Segmentation." This is a topic that is often studied in the asset pricing literature:

• The Effects of Market Segmentation and Investor Recognition on Asset Prices: Evidence from Foreign Stocks Listing in the United States
• The Effect of Market Segmentation and Illiquidity on Asset Prices: Evidence from Exchange Listings

It might be the case that some asset costs some amount in one country, but a different amount in another country, even after applying the appropriate exchange rates. Theoretically, this can happen due to market segmentation. Think about regulations that might prevent foreign investment, for example. So, anyway, you'll want to explore whether there are any regulatory barriers (or other things causes segmentation) that would affect your analysis.

A quote would be on an exchange, and currency involved is whatever is specified on the exchange. There may be multiple exchanges - with different prices. (Price deviations between markets for financial assets are normally small, but that depends on the exchanges being liquid.) Data providers translate those quotes to other currencies based on the appropriate exchange rate. You need to look at the exchange to get the details.

It is possible in this case that data providers blend quotes from multiple exchanges, and the exchanges could use multiple base currencies. You would need to dig into the details of your data provider to find out.

If you want to actually trade and the exchange is in another currency, you will need to factor in the exchange rate transaction, which have wider bid/ask spreads than wholesale exchange rate data that is normally quoted (unless you are a big customer).

It's not clear what level of answer you're looking for, so here is a much more basic answer.

There are indeed many exchanges with many different prices. However, if you have noticed that you could make money by exchanging your BTC for USD, exchanging the USD for SEK, and then exchanging the SEK for slightly more BTC than you started with - then someone else probably has noticed that too, and they'll make free money with the manoeuvre just as you will.

As people do this, the prices gradually equalise: you're making BTC more scarce relative to SEK so are driving its price up in SEK (making the last hop less profitable), and you're making BTC more common relative to USD so are driving its price down in USD (making the first hop less profitable).

So it is theoretically possible for such an opportunity for arbitrage to exist. In practice, however, in such a simple case, there are powerful forces acting constantly to equalise the price. As long as it's easy and low-fee to make all three transactions in the chain above (that is, as long as the respective markets are sufficiently liquid), people will make those transactions and the opportunity will naturally disappear.

If you identified a really unobvious opportunity for arbitrage (so unobvious that nobody else saw it before you) and the markets were liquid enough, then you could make a lot of money off it by exactly this manoeuvre!

In fact, very early on, the Bitcoin markets were absolutely full of arbitrage opportunities. The big players were not keen to trade in such risky places (it was routine for an exchange to just close down and run away with large amounts of other people's bitcoin - see the middle point on Forbes's article about this), so there was not such a strong force equalising the prices. As it's become more mainstream, though, more people started taking the free money, and the opportunities have dried up somewhat. So you may very well find that earlier in your dataset, the markets are much more obviously distinct.

The title says this question is about "financial markets". The body of text does not contradict that although Bitcoin is the only example discussed. If the question is about financial markets then the answer is that there are examples of markets segmented by currency.

For example Cameco is 20.23 CAD and 16.16 USD as of 4PM today. On Yahoo the symbols are CCO.TO and CCJ. I don't know what the currency rate was at 4PM but I'm sure that there are many times during the day when you can buy it cheaper in one market than the other based on internet quoted rates and prices. In reality the rate you can have might not be the same as what you see on Yahoo because there is a bid price and an ask price in a real market.

It is easy to name Canadian examples... Shopify, Bank of Montreal, Magna, Sun Life Financial, Open Text.

Edit: I think at 4PM CADUSD which means USD per CAD was 0.7994 which means that CCO.TO was 16.17 USD so it was cheaper to buy it in the U.S. market because you would save a penny.