The damage die of a psychic blade could increase at DM's discretion.
As you mentioned, the psychic blade fully qualifies as a monk special weapon and could benefit from martial arts and other monk features that require a monk weapon.
The damage die of the second attack could be modified by Martial Arts at DM's discretion following the rule Specific Beats General:
If a specific rule contradicts a general rule, the specific rule wins.
Both Martial Arts and Psychic Blades present some specific rules.
I wouldd (personally) rule that Martial Arts could also increase the damage die of this second attack, since the second psychic blade still is a simple melee weapon that is not heavy nor two-handed, and you also have to invest at least 5 levels in monk to increase its damage die, so it wouldn't break anything to allow it.
Other DMs might see the damage die reduction as a more specific rule, disallowing the second attack's damage die from being modified this way. This is also a fine way to rule it.
However, reading the feature RAW, you cannot create psychic blades more than once per Attack action, and you can create one as a bonus action only in specific circumstances.
Whenever you take the Attack action, you can manifest a psychic blade from your free hand and make the attack with that blade. [...] The blade vanishes immediately after it hits or misses its target [...] After you attack with the blade, you can make a melee or ranged weapon attack with a second psychic blade as a bonus action on the same turn, provided your other hand is free to create it. [...]
The feature lets you create one psychic blade when you take the Attack action, not when you attack as part of the Attack action, so features like Extra Attack cannot use more than one psychic blade, and it has been established with this question that this bonus action is not the same as using Two-Weapon Fighting (and Martial Arts by the same logic). The special bonus action of this feature allows you to do something similar to two-weapon fighting or using the bonus action attack of martial arts.