I don’t know about “xiao” or small, but in explaining “yen” it is perfectly possible to say “yen zi”. In fact, that’s what I would do. Actually I would transliterate it as “yan zi”, but you would probably read that as “yan” as in “yan can cook”, when it sounds more like how you would pronounce “yen”.
Normally, “yan” alone could mean a lot of things, including the bird, but “yan zi” is more specific - it means the bird. So it’s perfectly normal to say “yan zi de yan?”, which when translated would mean “swallow as in swallow?”. That of course makes no sense, but there you go. At least it’s better than saying “examine as in swallow”? (“yan” can also be translated as “test” or “examine”. Or rather, it sounds exactly the same as the word for test or examine.)
“yan” can mean bird. “yan zi” also means bird. But when explaining exactly which “yan” you’re talking about, “yan zi” enables the listener to identify which “yan” we’re talking about.
Perhaps a more helpful, although less authentic, translation would go something like
“My name is Yen Ling”.
“Yen as in yen zi, the swallow?”
“Yes, that’s right. Yen as in yen zi”
Of course, then people would pick on swallow being repeated. Or you could try “Yen as in yen zi (i.e. swallow)”?
To my ears, though, that passage is a perfectly normal way of introducing yourself and which characters make up your name. For example, when I’m telling people my chinese name, I would go something like
“My name is build strong. Build as in construction, **strong **as in strong healthy”.
Literally translated and bolding the equivalent bits, it would read something like
“My name is jian qiang, jian as in jian zhu, qiang as in qiang zhuang.”
If I simply said “jian qiang”, people might think the “jian” was the iconograph for “healthy” (understandable, as the sound is generally translated as strong with undertones of healthy, “jian” as in “jian kang”, which is better translated as “in the pink of health”)
… you know, this is a lot simpler when you have a chinese name.