Yes, positions are assets held under your name. (Saxo does not enter you in the share register, but that only affects you if you want physical dividends. Such as a watch or chocolate.)
If it is listed in position and there is a icon that says EQ (for Equities), yes you own stocks.
- EQ = Stocks
- FU = Futures
- BO = Bonds
- etc.
See “Platform settings” → “Trading Products”. There you can see what you are allowed to buy. I only buy stocks and ETFs, so I can’t comment on the other products.