In D&D Beyond, when I create a Shadow Kai, the creature type states that "You are Humanoid, You are also considered an elf for any prerequisite or effect that requires you to be an elf".

When I attempt to modify the racial stat bonuses, I get +2 to any stat and also +1 to another stat so I choose +2 to Strength & +1 to Dex.  This seems to contradict what the above statement says about the prerequisite, as an Elf race gets +2 to dexterity.  When I choose the extra stats, I also get the ability to teleport via Blessing of the Raven Queen.

Should the Blessing of the Raven Queen not be removed as I'm choosing the racial stats of the Elf?

You are reading this incorrectly.  The "count as an elf" thing does not mean you get to use the racial stats of an elf--it means you can take things that require you to be an elf, like Elven Accuracy, even though you did not choose "elf" during race selection.  You still maintain all Shadar Kai traits--those are static to the race itself.

That statement isn’t a prerequisite, just a descriptor that says you count as an elf for anything that requires you to be an elf like racial feats or magic items.

When you take the +2 Str & +1 Dex, that’s not in addition to the +2 Dex for being an Elf, but instead of it. You cannot double dip on the stats by taking both. So yes, you get the Blessings of the Raven Queen too.

So, to get +2 &+1 stats (+3 total)  and also get teleport is correct for a shadar kai?

Quote from craigjwhitmore >>

So, to get +2 &+1 stats (+3 total)  and also get teleport is correct for a shadar kai?

That is correct.

I've been playing with a DM who is of the opinion that I either can have +2 dex & teleport, or I can +3 without teleport, but not both +3 & teleport.

Quote from craigjwhitmore >>

I've been playing with a DM who is of the opinion that I either can have +2 dex & teleport, or I can +3 without teleport, but not both +3 & teleport.

 
Your DM is incorrect. When you play a Shadar Kai, you get all of its features: Ability Score Increases (+2&+1 or +1+1+1), Languages, Blessings of the Raven Queen, Darkvision, Fey Ancestry, Keen Senses, Necrotic Resistance, and Trance. You get the whole package.

It seems I can put all three stat bonuses to the same stat with Shadar Kai. I'd imagine this is a bug? Regardless of whether I pick +2 and +1 or +1 to three stats, I can put them all into Dex for example which seems broken?

I am playing a Shadar-kai cleric and put all 3 stat bonuses into Wisdom. It's up to you where you put the bonuses. 

Its a limitation of the system.

From the relevant section of TCOE:

Ability Score Increases

The Ability Score Increase trait in a race reflects an archetypal bit of excellence in the adventurers of this kind in D&D’s past. For example, if you’re a dwarf, your Constitution increases by 2, because dwarf heroes in D&D are often exceptionally tough. This increase doesn’t apply to every dwarf, just to dwarf adventurers, and it exists to reinforce an archetype. That reinforcement is appropriate if you want to lean into the archetype, but it’s unhelpful if your character doesn’t conform to the archetype.

If you’d like your character to follow their own path, you may alter your Ability Score Increase trait and assign ability score increases tailored to your character. Here’s how to do it: take any ability score increase you gain in your race or subrace and apply it to an ability score of your choice. If you gain more than one increase, you can’t apply those increases to the same ability score, and you can’t increase a score above 20.

For example, if the Ability Score Increase trait of your race or subrace increases your Constitution by 2 and your Wisdom by 1, you could instead increase your Intelligence by 2 and your Charisma by 1.

The website lets you do it because it's not been programmed in to restrict you.