Here are the problems/issues:
Most stars are born in clusters/associations but a cursory investigation of cluster demographics with age reveals that the vast majority of clusters do not survive to old age. The majority either are never gravitationally bound to begin with or become unbound in the first 10 Myr.
The Sun was likely born in a cluster of $10^3-10^4$ stars (Adams 2010). If it dispersed more than 4 billion years ago, then its members would have had time to spread all around the galaxy by now. That is because although the members would be kinematically coherent to begin with, the velocity distribution of stars in the Galactic disc becomes "heated" as stars scatter in the potential of giant molecular clouds and pass through spiral arms (Wielen 1977). They attain a significant velocity dispersion which means they could now occupy positions over a broad swathe of the Galactic disc and be displaced above/below the disc by $\sim 200$ pc. If we argue that the Sun's siblings could be in an annulus, with width $\pm 1$ kpc and thickness 400 pc, then if the solar cluster was $10^4$ stars, then the nearest sibling is expected to be at a distance of 160 pc.
Another way of looking at this: The density of stars in the Galactic disc is around 0.1 per cubic parsec compared to the expected density of solar siblings of $2.4\times 10^{-7}$ per cubic parsec.
Thus only about 1 in 400,000 of the local stars might be a solar sibling.
How could they be found?
Cluster members would have a similar age and a similar chemical composition.
We know the age of the Sun precisely and accurately. We don't have that information for any other star. Age estimates of field stars are imprecise, model-dependent and are of indeterminate accuracy. In the best cases - solar type stars and a little more massive - asteroseismology and the HR diagram position might give an age to about $\pm 0.5$ Gyr. But you need good data to do that and we don't have asteroseismology for the nearest 400,000 solar-type stars (in order to find ONE solar sibling). Even if we did, it would only narrow the candidates down by a factor of 10.
What about chemical abundances? There are now big spectroscopic surveys that have observed large-ish numbers of stars. Again, the most robust data comes for stars like the Sun, so that a differential abundance analysis can be done. The more numerous cooler stars have spectra which are more difficult to deal with and there may be systematic abundance errors when comparing with the Sun. Giant stars are bright, with narrow spectral lines, which is good, but their abundances may have been modified during their evolution. Thus we are limited to looking at solar-type field stars. No spectroscopic surveys (yet) have detailed spectra of 400,000 potential solar siblings.
Chemical abundance may also not be that discriminatory. In the solar neighbourhood, stars have an abundance dispersion of about a factor of two , centered quite close (a bit below) the solar metallicity. Good quality spectroscopic analysis can give the metallicity to 10%, so enough to resolve the distribution, but it would only whittle down the candidates by factors of a few.
Very high quality spectra could look at the detailed abundance mixture, but is only available perhaps for a few thousand stars. The majority also have an abundance mixture that is not very different to the Sun. Thus whilst the evidence is that the detailed abundance dispersion in a cluster is much smaller than the dispersion in field stars (Paulson et al. 2003), it does not look like the Sun is that unusual or has any unique chemical "markers" (Bensby et al. 2014). Some hope may be offered by the detailed mix of s- and r- process elements; the former may be age sensitive (e.g. Jofre et al. 2020), while the latter might reveal some peculiarity associated with contamination by nearby supernovae in the Sun's natal environment. At the moment, although candidate solar twins can be found, it is unclear by how much that narrows down the one in 400,000 figure, especially when you consider that abundance peculiarities may also be imprinted by later accretion events or some process associated with planet formation (e.g. Melendez et al. 2012) or that the abundance dispersion in a cluster may not be exactly zero (e.g. Liu et al. 2016).
In summary, the space density of solar siblings is likely to be very low compared with unassociated field stars. The properties of those siblings are not that unusual compared with typical field stars and we lack precise enough measurements of age and chemical composition to have anything more than a list of candidate solar twins at the moment. It's like looking for a needle in a huge pile of other needles.
