Aegis Climb does not reference Spider Climb and therefore does not ‘function like’ it.
But mostly yes
Spider Climb does very little that a Climb Speed does not.
- +8 to Climb checks (both)
- Can Take 10 on Climb checks (both)
- Retain Dex to AC while Climbing (both)
- Cannot use the Run Action (both)
- Requires the use of both hands (both)
- Can Climb across surfaces such as ceilings without handholds (Spider Climb)
Essentially, Aegis still need something to grab on to, but the option allows them to move more quickly and without sacrificing Dex to AC (protecting them from rogue ambushes and the like).