Had the same problem with the teak seatings in cockpit.
Softening of caulking was caused by (too extensive) application of teak oil. It seems, one has to decide between care for teak surfaces or care for caulking.
No way to get existing, softened caulking hardened again than to remove and replace it.
We took a cutting knife for our Fein-Multimaster tool
https://www.svb24.com/en/fein-cutting-blade-4-mm.html to remove the caulking and after this a decent sanding of the seams to remove all rubber remnants.
To avoid the need of sanding the surfaces,, we used masking tape on the teak surfaces and after this preparation we applied TDS 440.
Since we did this repair, we applied teak oil once a year in spring only sparingly (only one thin coat) and removed all excess amount with a soft cloth after several minutes.
Since then there was no further softening caulking.