Can we claim multiple home planets?

From this article, yes you can claim multiple home bases/planets, but you will have to start over each time. You can't have multiple at one time.

Base building sounds great, letting you claim a home planet and build an uninhabited base into your own outpost – you’ll have to start over if you find a prettier planet and want to build a base there instead.

As suggested from @warface I copy & paste my comment as new answer:

With the newest NEXT update (1.53) you can build several bases (on the same or on different planets). But you always start from scratch with only the base computer. Then you have to build all the needed stuff again (Refiner, Teleporter, Buildings...).

The only exception seems to be the big storage container. You have to build it again too, but your content is available immediately and is always the same in every new storage container. At least at my actual level where I can have only 1 storage container.

And just for the record: You can fill this container from everywhere, but you have to travel back to (one of) your storage container(s) to pickup some items.