Why is there a discrepancy between the underlying meaning of variance between basic statistics and meta-analyses? Normally, I would say that variance (and, in extension, standard deviation since it's just the square root of the variance) refers to a measure of deviation that describes how data points within a sample varies. If you, on the other hand, want to say something about where the sample means are expected to lie within a population (that is, you want to inference something rather than just describing something), you extend this concept by using the sample size and derive the standard error. More specifically, the standard error (SE) is computed by
$$SE=\frac{sd}{\sqrt(n)}$$
where sd stands for the (sample) standard deviation and n stands for the number of whatever your individual measuring points are (for example, research subjects).
When doing meta-analyses, you normally talk about a data points effect size (which, for example, can be a standardized difference such as Cohen's d where two groups are compared) along with it's variance, where the variance is calculated by taking the sample size of the data point into account, that is, contrary to the concepts above, it's a kind of inference. This can, for example, be calculated using the mes function of the compute.es package, where you get Cohen's d as well as the variance of Cohen's d, and these measures are subsequently used, for example, in the rma function of the metafor package. More specific, the variance of Cohen's d, $v_d$ is computed by
$$v_{d}=\frac{n_1+n_2}{n_1*n_2}+\frac{d^2}{2(n_1+n_2)}$$
where $n_1$ and $n_2$ are the number of individual measurements of group 1 and group 2 respectively.
Now, when reading up on network meta-analyses, the netmeta function of the netmeta package, wants you to supply standard error of your effect sizes rather than variances of your effect sizes, which I'm used to.
Why is there a discrepancy between how these concepts are used when it comes to meta-analyses and when it comes to more basic statistics (or is it me that have misunderstood something)? That is, it's calculated without taking the sample size into consideration in one context and with the sample size accounted for in the other. And what is meant by the standard error in this meta-analytic context and how do you calculate it?

What you denote as $v_d$ is not the variance of raw data, but a variance of a derived statistic based on a bunch of raw data. To distinguish the sample variance of raw data from the variance of a statistic, one can describe the latter as sampling variance, that is the variance in the statistic that you would obtain under repeated sampling of new study units under identical circumstances.
The standard error of a statistic is the square root of its sampling variance. Hence, just as there is not one equation for computing the sampling variance of a statistic (every statistic has its own equation for computing/estimating its sampling variance), each statistic has its own equation for the standard error. So, if you take the square root of $v_d$, you have the equation for computing/estimating the standard error of a standardized mean difference.
