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
Very Unlikely, but...
There is not clear RAW that indicates how this should work one way or another, but here's some food for thought.
Infusions only work on Objects...
The Steel Defender is a Creature, not an Object.
Its 'Natural Armor' is presumably its metal 'skin.' There is no precedent in D&D 5E for part of a creature being considered an Object. Thus, while the RAW doesn't specifically say you can't enchant the "skin" of a Steel Defender, its skin is not an object--it's part of the creature. And, frankly, a lot of creatures have "Natural Armor" in their statblock--if you used that as your justification to be able to enchant it, then you'd also be able to enchant the skin of a wolf.
Armor Infusions specify "A Suit of Armor"
As noted here
Metal skin is not a suit of armor. Enhanced Defense can only be applied to a Suit of Armor, not a Creature that is Armored. Naturally, this applies to other things as well--weapon infusions must target an actual weapon--not part of a creature.
But Barding is a thing.
Your Steel Defender may have the form of an animal--there are already rules for putting Armor on an animal.
If your Steel Defender is shaped like a panther, and you could buy barding for a panther...then you can buy barding for your Steel Defender. Once you've put barding on it, you can enchant the barding because it's a suit of armor.
Beyond this, your Steel Defender could be humanoid
If you decide that it has a humanoid appearance, you might be able to just stick normal armor on it.
Other Infusions
The rules for non-humanoid Creatures using magic items are...
So while you cannot enchant the Steel Defender itself, you can equip it with things you make. Keep in mind...
So if the item you give it (such as a weapon) would require an Action other than the ones listed to use--it can't use it. However, if you throw a Cloak of Protection over its back, give it a Belt of Hill Giant Strength and an Amulet of Health...all of those magic items will work on it just fine.