Toolkit: Assets are not re-built when a subsequent `synth`/`deploy` is called within the same node:process

### Describe the bug

Now with `toolkit:deploy` and `toolkit:watch`, customers can write programs/libraries to invoke `deploy`/`synth` multiple times within the same process. This however causes some of the assets to not be rebuilt causing the change to be missed from the deployments.

### Regression Issue

- [ ] Select this option if this issue appears to be a regression.

### Last Known Working CDK Version

_No response_

### Expected Behavior

Assets should be correctly rebuilt for every changed asset for every deployment/synth

### Current Behavior

some of the assets are not rebuilt

### Reproduction Steps

Use this app https://github.com/Amplifiyer/cdk-synth-repro-app and run `npx tsx main.ts` and observe that the second deployment at https://github.com/Amplifiyer/cdk-synth-repro-app/blob/deefb326c63b918b76f5ced37ded038d4a4e8a5e/main.ts#L44-L47 does not pick up the change introduced.

### Possible Solution

Workaround is to have customers call `AssetStaging.clearAssetHashCache();` before each invocation.

A correct fix would be for this to be handled seamlessly at the CDK end. This may require AssetCache to be changed from singleton to an instance.

### Additional Information/Context

_No response_

### CDK CLI Version

0.1.5

### Framework Version

_No response_

### Node.js Version

20

### OS

MacOS

### Language

TypeScript

### Language Version

_No response_

### Other information

_No response_