| Tvarog |
Strictly RAW, it looks like it does. Since it does specify "attached", it probably can't actually move the two of you, but it should be able to take move equivalent actions (and standard, etc) just like any non-tumor familiar.
PFSRD entry on Tumor Familiar wrote:
The tumor acts as the alchemist’s familiar whether attached or separated (providing a skill bonus, the Alertness feat, and so on).
PFSRD entry on Familiar (Wizard class ability) wrote:
Small-sized familiars threaten the areas around them like Small creatures, and can be used to flank enemies, though both familiars and their masters are often loath to use such tactics, as the result is often a dead familiar.