Just to add to the above responses:
I fully agree with response #3 above.
I did this in August of 2019.
I was arriving to Southampton on QE, then leaving on QM2 four days later.
I was waiting for some notice to appear in the daily program about this,
but I should have gone to the Purser's desk earlier in the trip.
The cost was $65 per suitcase (which was billed on the first trip, you could use OBC).
I believe I received a form to fill out, along with special color luggage tags.
When disembarking, I was required to go to the area where the bags were,
just to verify my bags were in the group (which they were).
The next time I saw the bags was in my cabin when I boarded QM2 four days later.
I would definitely do this again.
BTW it was called transfer, not storage