As broadcast, the two are completely independent universes. However, there is strong evidence that the original intent was for them to be sequential. With, what became Andromeda, showing a universe after the fall of the Federation. Looking at it from that point of view, you can definitely see the similarities. Humanity joined the Commonwealth and the Federation was founded at, essentially, the same point in Earth's history. Each of the races in Andromeda parallel the ones in Trek:
Vedrans - Vulcans or maybe the precursor race
Humanity - Humanity
Nightsiders - Ferengi
Nietzscheans - Advanced Klingons
Magog - Borg (In reverse. Ultra religious vs Cybernetic)
Often times, these races were done in a bit of parody, I believe. Since the iconic traits of each race were exaggerated in Andromeda as verses Trek.
The technology in Andromeda is a legitimate outgrowth of that seen in TNG. In the TNG era, the Federation was just beginning to experiment with wormhole or slipstream technology. Andromeda shows that technology perfected.
The ship design is radically different though. Trek took a very scientific and utilitarian view of ship design. Andromeda uses a much more organic ship design. Also, Andromeda's ships of the line are designed primarily for combat. This could be a logical outgrowth of a Federation that had become fascistic and conquest oriented. We see the beginnings of that movement during the TNG era, especially in DS9. The Dominion War era was the end of the Federation's innocence.
Also, the use of highly evolved AIs has early parallels in Trek with Data and Lore. The Picard series has really rounded that out and brought AIs, or Synths as they are called in Picard, in to being as fully rounded individuals. Another thousand or two years of evolution of the technology may result in Andromeda style AIs capable of operating entire warships on their own.
Also, the concept of the Commonwealth itself is a direct parallel to the Federation. In fact, the new Commonwealth that Hunt tries to build throughout the series is actually a federation of worlds, in terms of governmental system.
One stark difference between the shows is genetic manipulation. In Trek, the practice was essentially banned after the eugenics wars, excepting for medical necessity. However, in the Andromeda universe, genetic manipulation is nearly universal. So much so, that almost all the main characters are somehow genetically enhanced and there is an entire race that has been enhanced and pursues genetic perfection.
So, there are plenty of parallels and equivalencies between the two series. Enough to make you believe that the two are mirrors of each other. However, there is no crossover between the two. And, while many believe it was Roddenberry's intent to sequence 'Andromeda' after the fall of the Federation, this could also be a case of a single showrunner and writer recycling base concepts in new ways.