No, they're not (at least as of the current UA).

The UA Way of the Astral Self monk's Arms of the Astral Self feature says, in full:

At 3rd level, your mastery of your ki allows you to summon a portion of your astral self. On your turn, you can spend 2 ki points as a bonus action to summon the arms of your astral self for 10 minutes. These spectral arms hover near your shoulders. You determine the arms’ appearance based on the qualities of your character.

While your astral arms are summoned, you gain the following benefits:

• You can use your Wisdom modifier in place of your Strength modifierwhen making Strength checks and Strength saving throws.
• The arms are monk weapons and have a reach of 10 feet. The arms dealradiant or necrotic damage (your choice). When you attack with thearms, you can use your Wisdom modifier instead of your Strength orDexterity modifier for the attack and damage rolls.
• Immediately after you use the Attack action with your astral arms onyour turn, you can make one extra attack with your astral arms as abonus action. The number of extra attacks increases when you reachcertain levels in this class, increasing to two at 11th level andthree at 17th level.

No part of the description of that feature (or any other feature of the subclass) says that attacks with your astral arms count as unarmed strikes. In fact, unarmed strikes aren't even mentioned in the UA. Furthermore, it says they are monk weapons; if they counted as unarmed strikes, this would not make much sense, as the Martial Arts feature describes unarmed strikes and monk weapons separately and distinguishes the two.

This is just an error in D&D Beyond's implementation of this UA. I reported the error in the feedback channel of their official Discord server, and a mod on the DDB Discord (Stormknight#9773) replied to my report with the following explanation:

With regards the Astral Monk - the attacks are modelled as unarmed strikes on D&D Beyond, as it would take additional development time to introduce a modifier for innate attacks that count as monk weapons.

With Unearthed Arcana, we generally don't feel it is worth diverting people from feature development to work on implementing playtest content, where we already have a close functional fit.

As always, the rules stated under the actual ability supercede the meta-data attached to the ability.

If/when the subclass is included in a published sourcebook, we will absolutely take the time to ensure it is 100% correct.