My martial arts sensei walks like this. Smooth as a cat. He has done Ninjutsu and traditional Jujutsu training in Japan.

See also https://en.wikipedia.org/wiki/Across_the_Nightingale_Floor

But it's partly out of taking short steps - to maintain balance. If your feet are too far apart you're vulnerable to tripping and leg sweeps and getting unbalanced by being hit in general.

I find it impossible to do toe first if my step is the usual length.

Same, I noticed this years ago. I'm like a ninja walking around the house. The banging around of heavy feet drives me nuts, my parents are like this.

I've walked like that for all my life (I would stand on my toes a lot), so you saying it's not abnormal when walking around the house makes me feel better.

I've moved to a house with hard / laminate flooring recently though, and it took some time to adjust - even with my weird gait the floor was uncomfortable to stand on for extended periods of time without shoes.