Your other weapon also needs to be light (or have the duel wielder feat) in order to qualify for duel wielding a Shadow Blade.
Pole Arm Master works though.
Hu?
When you use your pact of the blade feature it becomes a magical version of the weapon you choose, so if you want to duel wield you can choose a shortsword and it will be considered light and finesse.
the shadow blade is finesse and light, and thrown properties (range 20/60) per the spell so it qualifies as an off hand.
This was brought up in response to "discount two weapon fighting" which adds your ability score to the damage of your off hand attack:
Yes, you can make your Hex Warrior weapon different from your Pact of the Blade weapon and use charisma for both.
In fact, it may be possible to get a discount Two Weapon Fighting Style if Hex Warrior gives you charisma to damage on your off-hand.
So shadowblade used as and off hand short sword gets 2d8 instead of 1d6 an average of 6 damage more as a level 2 spell which is roughly equivalent to 1d6+5 from 20 charisma and hexwarrior as the post suggests. It also gets advantage on attacks in dim light and darkness so if your strength and dex are both 10 you can still get and effective +5 to hit, and if your charisma is 20 your likely higher level and your pact spell slot will automatically increase, increasing the damage of shadowblade in turn. With only an investment in one spell selection you already get and any amount of strength or dex you can duel wield with better than a "discount two weapon fighting" on top of that it can't benifit from hexwarrior and it is a warlock spell so you don't want to use it in your main hand since it is exactly the same in your off hand.
So basically any pact of the blade warlock can use shadow blade as a "discount two weapon fighting" the only difference with the hexblade is that the hexwarrior feature lets them us CHA for their main hand, but with shadowblades advantage to hit in dim light and darkness and greater damage damage even a +2 dex investment for descent AC with medium armor your off hand is actually better in caves and dungeons than your main hand even with hexworrior.
If your saying your taking a damage reduction for a light main hand... the most damage from a single weapon is 2d6 +Modifier we will call if 5 average 12, Main hand is 1d6+modifier and off hand is 2d8 average of 17,5. If you counter with thirsty blade your still getting that with duel wielding but also your now at least level 5 so your shadowblade now does 3d8. Thirsty blade only works with you pact weapon so their is no advantage to putting shadowblade in your main hand. Damage wise thirsty blade with a short sword and level 3 shadowblade is a possible 2d6+5+3d8 (30.5) vs 2d10+10 +1d4+5 (23) both form 3 attack and shadowblade will increase by another 1d8.
The only real down side of shadowblade as an off hand weapon for a hexblade pact of the blade warlock is that its concentration, however its also continual for a minute and warlocks can rely on having each battle due to pact magic. They can re-cast it as a bonus action but with some investment in con saves / warcaster its relyable enough.
i am not saying that is "the best" i just saying its a reasonable option with a buy in as cheep as one spell selection that's reasonable for any pact of the blade warlock.