Yes. Your steel defender can attack, so long as it's acting independently.
From the Controlling a Mount entry..
An independent mount retains its place in the initiative order. Bearing a rider puts no restrictions on the actions the mount can take, and it moves and acts as it wishes.
And then from the Steel Defender entry, it reads..
It is friendly to you and your companions, and it obeys your commands. .. It can move and use its reaction on its own, but the only action it takes on its turn is the Dodge action, unless you take a bonus action on your turn to command it.."
Given this, when your steel defender is acting independently, it's obeying your commands. There is no indication that commanding your steel defender and controlling your mount are the same thing.
You can control a mount only if it has been trained to accept a rider. Domesticated horses, donkeys, and similar creatures are assumed to have such training. The initiative of a controlled mount changes to match yours when you mount it.
And here is why. A steel defender hasn't been trained to know that you want it to go faster when you kick its sides, or that you want it to right or left by pulling on the reigns. The steel defender just obeys your commands, which is separate and distinct from controlling a mount.
There is also no mechanical benefit to controlling it; its initiative is already the same as yours, and it already moves as you direct you, and those are the only benefits of controlling a mount. Controlling it would only limit the actions it could take.
I heavily edited my original post because, after thinking about it, I was kind of all over the place and wanted to give a more concise, sensible answer.
Yes, if you want
The rule for the Steel Defender states:
You determine the creature's appearance and whether it has two legs or four; your choice has no effect on its game statistics.
The wording is sufficiently vague that you could say "My steel defender has reconfigurable limbs, allowing me to change it between 2 legs and 4."
The rules do not say that you determine the Steel Defender's appearance when you create it. Obviously, this does happen, but the wording permits the interpretation that one can choose to alter its appearance subsequently because no timing is specified.
And why should they? You might also paint it or glue a wig on it. These are cosmetic changes permitted by the fact you're playing an open ended RPG and do not violate any explicit rule.
Since your DM is so "by the book," it's worth remembering that Rule zero is in the book and grants wide leeway to DMs and players to act as they want.
Given the fact that we're dealing with a cosmetic or narrative decision, you have plenty of (ahem) leg to stand on.
Best Answer
Yes, they can
The Steel Defender feature says
There is nothing in there limiting the actions to involve only opponents you can see. It can be any action.
Obviously, to command it to attack a specific creature, you need to be aware that the creature is present. But a creature you cannot see is not automatically hidden from you.
Note however, that the Steel Defender's Force Empowered Rend attack has a limitation:
"You" in the defender's stat block refers to the caster (for example, it has hit points of "2 + your Intelligence modifier + 5 times your Artificer's level" and the defender does not have any artificer level). The defender itself is referred to as "the defender" in the stat block, as in the Defense Attack reaction ("The defender imposes disadvantage […]").
That means, while you can command it to attack the creature, it cannot carry out the Force Empowered Rend to do that, as that works only on a creature you can see.