I offer my 2 cents:
I've come across several ID firms with arrangement like these. Some of the reasons are: ID business just an arm of a parent company and HDB license under the name of parent company, ID company has several sister company to spread out businesses so as to avoid GST (GST not mandatory if turnover less than S$1m), company too new to qualify for HDB license...
With regards to "Can I still claim if anything happens during reno stage???" In my opinion, homeowners are not likely to claim anything "if anything happens", whether ID firm is HDB registered or not.
HDB license only comes into play when contractors breach HDB renovation rules/guidelines, eg. damage to building structure, carrying out work outside permissible timing, etc. These are usually NOT the cause of disputes between IDs/contractors and homeowners. Most contractors has no problem adhering to HDB rules.
Issues such as, delay of reno progress, promised work not delivered or not up to expectation, really bad workmanship, damage to properties (excluding building structure), etc. are those which cause homeowners most grievances, and which unfortunately HDB offers no aid at all.
I think the most HDB would do is to revoke the offending ID company's HDB license if there are too many serious complaints lodged against it with the authorities. However, this is not really a consolation to homeowners at all, cos the unethical businessman can set up another new company and continue bad business practice.