Amino only covers the next step; growing your bacteria. The kit comes with a single, "living nightlight", sample that you can grow. They also plan to offer additional samples ("Apps"...) that you can buy. As far as tinkering, it sounds like the only intended modification is to the growth liquid: grow in water vs grow in liquid-X.

The company they're working with, Synbiota, does have a "DNA Tinker Studio" kit that looks interesting. It's not full synthesis either though. It contains specific set of available 'parts' to work with, that you combine and get existing bacteria to take up.

There are services, not difficult to find, that offer actual digital->DNA synthesis. Current state, they are expensive, and are focused-on/able-to create only relatively short sequences. Sequence->Modify->Synthesize->Grow, or even targeted splicing, aren't really available things yet, AFAIK.

Here here- this potentially offers a huge amount but is vague enough to offer all the promise of a childhood chemistry kit bundled with the realisation on unwrapping that you've been sold up the river with overhyped claims. How hard would it be to generate your own dms sewuences? Could you order genes with appropriate restriction end points and insert that? Does the kit actually perform all transformation and selection automatically?