I would like to know what is the closest separation achieved by two satellites.
Let's restrict it to
This answer state that it would be "Few kilometres"
Whereas ESA (non commercial entity) achieved 4km
There's a disadvantage to placing satellites too close together for no reason, if struck by a micrometeoroid pieces of both the space debris and satellite pose a danger to its companion; it's essentially doubling the chance of a strike. Here's what happens when a bullet strikes a sheet of glass:
The answer you linked to for "How closely spaced are satellites at GEO?" says: "... we can calculate that they were during those two time periods at the minimum distance to each other at roughly 1.1 km (0.666 mi).". This is further discussed in user TidalWave's question: At which point would two GSO/GEO satellites with similar orbital elements be closest to each other?
GRACE-FO used two satellites that orbited about 137 miles (220 km) apart. They were able to measure their separation distance to within 10 micrometer, about the diameter of a blood cell, enabling them to sense subtle differences in Earth's gravity field from location to location. Flying at an altitude of more than 300 miles (500 km), they were able to detect gravitational differences on the planet's surface equivalent to that of a 300-km disk of water only one centimeter thick. Wikipedia's GRACE webpage.
In an upcoming mission planned for 2020 the ESA will launch PROBA-3 which will vary it's distance between 25 and 250 meters. That will be the closed deliberate distance ever. I'm not including tethered or physically connected satellites.
Other references:
ESA's Satellite Formation Flying webpage.
"Formation Flying: The Future of Remote Sensing from Space", by J. Leitner - 2004.
(.PDF) - "NASA's Autonomous Formation Flying Technology Demonstration, Earth Observing-1 (EO-1)", by D Folta - 2002.
Other distantly related info:
You can draw plots of satellite locations using software such as SaVi and the available SaVi satellite plot scripts. An example rendering of just the Iridium satellites, which are over 212 km apart.
Space debris is also a concern, if space debris (or other satellites) collide with a satellite it sprays shrapnel that greatly affects nearby satellites; this is a good reason to maintain reasonable spacing apart, even if maintaining precise spacing distance.
Wikipedia space debris webpage offers this image:
There's enough junk and man-made (functioning) objects to avoid that flying deliberately close is avoided. Doing so is almost always unnecessary. Discussions are underway for flying guide-star satellites around the James Webb Telescope, see the article: "Laser Guide Star for Large Segmented-aperture Space Telescopes. I. Implications for Terrestrial Exoplanet Detection and Observatory Stability" by E. S. Douglas et al. Published 2019 January 4, older no-paywall arXiv version.
How About Half a Kilometer
The GRACE mission was mentioned in a very complete answer by @Rob (mentioned as GRACE-FO, which is the current mission replacing GRACE). However, unmentioned is the fact that the satellites swapped positions periodically. Since the satellites use K-band (and laser for GFO) ranging, the rear satellite was exposing its K-band "horn" to bits of atmosphere at the low mission altitudes (in the 300-400 km range). This was expected to be a possible cause of instrument failure and thus the satellites (referred to as GRC-A and GRC-B) periodically swapped positions in orbit.
During this maneuver, the trailing satellite "overtakes" the leading satellite and both rotate to keep the K-band instruments facing each other. This is a quality summary of the maneuver done in December 2005 (poor web formatting but oh well), and it is noted that the satellites had a range of 406 meters at closest approach.
This link suggests that during a 2015 swap the range at closest approach was under 300 meters, but there's really not much more detailed information.
GRAIL was a GRACE-like mission in lunar orbit, which flew at a similar separation. Since there is no lunar atmosphere, the spacecraft did not swap positions to my knowledge. However, this did allow them to fly extremely low, producing excellent maps of the lunar gravity field.
GRACE-FO is currently flying, and was doing on-orbit checks during its early mission phase. It is quite possible that the two satellites came even closer during checks, but I can't find any reliable information on that.
If the intention is to get really close, rendezvous is possible. Though is common to just take them up together. I.e: LISA pathfinder has 3 separate bodies that have been held close for prolonged periods of time. Also: docking is a thing as is landing on asteroids, so if we want to get close, we can.
Otherwise there is really no need to have satellites close, but with significant relative velocity (other than anti-satellite weapons which are also a thing so if you allow this: 0, though it is arguable the weapon itself is not a satellite as it's sub-orbital in all cases I know of).
There is a different question: what is the closest satellites are permitted to get? This really depends on accuracy of the calculations and its really more a "what odds are acceptable" but there are regularly sub-kilometre closest approaches that could be but aren't corrected. Often when they are, 2km is considered "safe distance".
