Are unpaired samples always generated by independent random variables? I am confused by the terminology because the term "independent samples" is usually used as a synonym for "unpaired samples". Does this fact mean that unpaired samples are always generated by independent random variables (assuming we have a probability model for our data)?
In other words, let's assume that we have a probability model for our data, that is we have two random variables: $X \sim F_X$, $Y \sim F_Y$. These two random variables correspond to two populations. Next, we consider two i.i.d. random samples $(X_1, \ldots, X_n) \overset{\text{iid}}{\sim} F_X$, $(Y_1, \ldots, Y_m) \overset{\text{iid}}{\sim} F_Y$ where there is no one element of $(X_1,\ldots,X_n)$ such that we can "match" it with some element of $(Y_1, \ldots, Y_m)$. We call such samples unpaired samples (of random variables).
Is it true that $X_i$ and $Y_j$ are independent random variables, $\forall i,j$? Or, equivalently, is it true that $X$ and $Y$ are independent random variables?

P.S. It is well known that paired samples (often called dependent samples) can be generated either by dependent or by independent jointly distributed random variables $X$ and $Y$ (see examples 1 and 2 below). But my question above is about unpaired samples.
Example 1. We have $n$ random people and measure the same characteristic (for example, body weight) in two different moments of time, this will give us $n$ pairs of numbers $(x_1,y_1), \ldots, (x_n,y_n)$. Here we can treat these numbers as realization of two paired samples (of random variables) $(X_1,\ldots,X_n) \overset{\text{iid}}{\sim} F_X$ and $(Y_1,\ldots,Y_n)\overset{\text{iid}}{\sim} F_Y$, which were generated by two dependent jointly distributed random variables $X$ and $Y$.
Example 2. We have $n$ random people and measure two totally different, unrelated characteristics (for example, year of birth and gender), this gives us $n$ pairs of numbers $(x_1,y_1),…,(x_n,y_n)$. Here we can treat these numbers as realization of two paired samples (of random variables) $(X_1,\ldots,X_n) \overset{\text{iid}}{\sim} F_X$ and $(Y_1,\ldots,Y_n) \overset{\text{iid}}{\sim} F_Y$, which were generated by two independent jointly distributed random variables $X$ and $Y$.
If we have any doubts, we can use the chi-squared test of independence to find out if paired samples $(X_1,\ldots,X_n)$ and $(Y_1,\ldots,Y_n)$ are generated by dependent or independent jointly distributed random variables $X$ and $Y$.

Independence has a formal definition, i.e. $f_{X,Y}(x,y)=f_X(x)f_Y(y)$.
On the other hand, I do not think that the term 'unpaired samples' has a formal statistical definition, and, at the risk of sounding snobbish, it is not a term that I ever hear formally trained (bio)statisticians use. Non-statisticians I work with will use that term, and I encounter the phrase in medical journals; when I encounter it, I assume that the speaker who uses this term is referring to the manner in which the data were collected, i.e. that the measured variables were not matched when they were obtained. So, to me, the term only makes sense in the context of a study design.
However, if I take your question at face value, then I think the answer is 'no': if all you tell me is that $X$ and $Y$ are generated from marginal distributions $F_X$ and $F_Y$, respectively, and you don't tell me anything about the joint distribution of $X$ and $Y$, then I cannot tell you whether they are independent.
Stepping onto my soapbox
The 'unpaired t-test' is another misnomer and a term that I never hear statisticians use, despite the popularity of the term in the literature. Just call it a two-sample t-test.
Stepping off my soapbox
