Before we get started here, I want to preface this section by saying that if you have not completed the game, you will be spoiled if you read too far into this section. I am not only going to cover the meaning of his alias, but the meaning of his biological name as well. At any rate, while names in real life might not have too much significance, video game characters are usually given names for a reason, so it pays off to dig a little into the meaning! Here's what I've deduced about his various names.
Shadow is the name that we know him by throughout the entire game, so even though it's just an alias, it makes sense to look at it first. Let's take a look at the dictionary entry (it's quite long, so I've cut out a few redundant definitions):
1 : partial darkness or obscurity within a part of space from which rays from a source of light are cut off by an interposed opaque body
2a : an inseparable companion or follower; b : one (as a spy or detective) that shadows
3 : a source of gloom or unhappiness
4 : a state of ignominy or obscurity
I think it's pretty obvious what Squaresoft was going for here. The name Shadow is associated with a mysterious darkness that seems to define his character. Even when we think we've gained some insight into his past, we cannot be 100% sure of what happened. For example, we only learn of Shadow's past through dreams, not a direct account. There are many questions left unanswered, and thus his past is still left "partially dark and obscure." Plus, you know, his appearance is that of a Shadow -- his body is covered entirely in black, and the only visible feature is his eyes.
The name Shadow also fits his occupation quite well -- after all, a "shadow" is someone who follows, undetected. This term is usually associated with spies and detectives, so why not throw an assassin in there as well? Shadow always seems to be running into your party, which makes you wonder if he's been shadowing them all along...
Finally, there is the association with the phrase, a shadow of his former self. For me, this is the first association that comes to mind (even though it's not really from the definition). Shadow is someone who has endured a lot of personal tragedy, and as a result, he turned to a reckless life of killing, just to make ends meet. It was only fitting that he took on the alias "Shadow" to represent the extent to which he has declined as a person. But really, I think all three of these associations fit in with his character very well -- I don't know that he could have chosen a better alias for himself!
Shadow's real name is a bit more difficult to unpack because, well, it's not a word. However, we can look at its origins and a couple of associations it has.
Origin : Scottish, Welsh
Meaning : The name Clyde is thought to come from a river in Scotland -- the Clyde river. It is thought to mean either rocky eminence, heard from afar, or warm.
The origin of the name Clyde does not provide a lot of deep insight into Shadow -- part of that is due to the lack of a clear meaning behind the name. All we know for sure is that the name is derived from a major river in Scotland, which clearly has nothing to do with Shadow.
However, it's more than likely that Squaresoft selected this name due to its association with Clyde Barrow, part of the infamous duo, Bonnie & Clyde. This reference is quite obvious, seeing as how Clyde's partner's name, Baram, sounds similar both to Barrow and Bonnie. So what do Bonnie & Clyde have to do with Clyde and Baram? Well, Bonnie & Clyde were an infamous duo of outlaws who robbed banks (and other places) during the 1930s. They captured the attention of the press, but eventually the police caught up with them and killed them, near their Louisiana hideout. While Baram is not a woman and Clyde does not die in FFVI, the situation is nearly the same -- after continued success, Baram is finally caught and (presumably) killed, and Clyde narrowly escapes.
Assuming that Shadow really is Relm's father, his last name is Arrowny. This is not a common last name (in fact, when I searched for it in google, nothing came up except Shadow and Relm!) However, it sounds like it is of British origin, like the name Clyde, so it meshes well with his first name. I guess the developers just randomly came up with this name...