Yes, you can upcast spells using sorcery points.
First, Psionic Sorcery says:
When you cast any spell of 1st level or higher from your Psionic Spells feature, you can cast it by expending a spell slot as normal or by spending a number of sorcery points equal to the spell’s level.
The first notable phrase we observe is:
When you cast any spell of 1st level ...
Reading the feature more carefully and "Casting Spells at a Higher Level," it's clear that this is not possible:
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. For instance, if Umara casts magic missile using one of her 2nd-level slots, that magic ...
No further guidance is given, but it seems most likely yes
having the same characteristics or qualities as; similar to.
The rules on spellcasting archetypes offer no help for this question, so we are left with the rules you quote from Sorcerer and their archetype. Unfortunately, the phrase "cast spells like a sorcerer" is a bit ...
It cleary can be twinned. It has a range of 60ft, and can only target 1 creature. by what is clearly written on the rules of twin spell it can be twinned. and to be frank from a balancing perspective 5 points is a heavy investment for this