"That's something I can't understand though.. when people can't even remember their own songs and tunes! It seems so odd when they have actually written the things themselves."
And if it's a new song, there's no-one out there who can help you if you forget!

But it's not about whether you've written them yourself or not: it's about how your own personal memory works. I do not have a photographic memory and have always struggled to learn WORDS. (Not just since I got older!) I "know" loads of songs when someone else is leading them, but not nearly so good at leading a song myself, tho' gradually over the years I have increased the number of songs I CAN sing without words in front of me: but there are HUNDREDS more that I like to sing, have sung, know how they go, yet would still need the words with me, if only to glance at.
In my working life, I had to learn so many facts and figures that my poor little brain won't take any more!
I have more of an "audiographic memory" in that I remember tunes much better: it maybe doesn't always manifest itself as being able to play you a tune, tho' I could usually sing the melody to you. (But with my efforts at doing more "ear playing", even this is getting better!)