These are really two separate questions remotely related due to the fact they are based on information recorded in the camera's metadata. We'll look at each one separately.
While the lens works well in autofocusing, I found the focal length was incorrect in the EXIF data.
and
Can that tap-in console handle my situation?
If the cause of the incorrect EXIF Info is a known bug in the lens' firmware, then using the Tap-in Console application with the hardware USB Tap-in dock to update the lens' firmware should resolve the issue, assuming Tamron has released an updated firmware for the lens that corrects the bug in the current firmware running in your lens.
... why there's no serial number in the metadata?
Most often, when certain information is not displayed in metadata the way we expect, it is not because the information is not there. It is because the application used to view the metadata does not know what to do with certain fields in the 'Maker Notes" section of the EXIF Info. Manufacturers are free to use whatever methods they wish to record "Maker Notes' information. There are no "standardized" fields in the 'Maker Notes' sections and thus there are no "standardized" ways of recording whatever information the camera maker desires to include there.
This answer to How can I compare the lens serial number from the EXIF of a picture, to the serial on my lens? illustrates how the correct data can be included in EXIF Info (because some applications do correctly interpret/translate/display it) but some EXIF viewers do not accurately interpret/translate/display the included information.
Keep in mind that many camera and lens makers use a different "internal" serial number for a particular body or lens than the serial number printed on the exterior of the piece.