I think the main reason Garrus is almost universally liked is that he works well as a character regardless how Commander Shepard is played.
He compliments both Paragon and Renegade Shepards very well whereas some characters work much better with a particular type. I don't think Kaiden or Liara would be friends let alone lovers with a Renegade Shepard for instance.
The Femshep/Garrus romance is probably the best in the trilogy. Hell in my opinion it's one of the greatest love stories ever told.
A platonic relationship works as well while he provides a pretty compelling Bromance for ManShep.
In short, Garrus seems to fit as Shepard's wingman regardless of roleplaying.
The fact that he's around for 90% of the trilogy makes him a unique squad mate ( Tali is the only other squaddie to be in all three games but in both ME2 and ME3 she joins up later than Garrus).
Despite this abundant screen time the character isn't forced on the player though ( this seems to be a common complaint about Liara from some fans). Garrus is almost entirely optional. You don't have to recruit him in ME1 and while he's compulsory in ME2, once recruited you don't have to interact with him at all. In which case he will almost certainly die in the Suicide Mission and you wont see him at all in ME3.
It's strange because as far as I'm concerned he's the second most important character in the series yet it's possible to play the entire trilogy and Garrus will only be on screen for about ten minutes. Bizarrely such a major character (and in my own playthrough he is THE major character after Shepard) is utterly irrelevant to the plot.
I think this is quite important as Garrus' presence in the game is entirely on the player's terms.
He's also pretty handy in a fight. In ME2 and ME3 he has just the right combination of powers and weapons to handle any enemy.
I also think he's well written in all three games despite having a different writer each time. Walters, Weekes and Dumbrow all had their own take on Garrus but each built on the previous work. Garrus is different in each game but the change feels like genuine character development rather than inconsistent writing.
Last but not least there's Brandon Keener's excellent voice acting. That guy simply nails every single line.