Though a more precise variant that specifically means "lucky" exists (udachlivyi, meaning possessing greater chances), schastlivyi is more often used to describe a person, with the semantic content being that they are both happy and lucky.
For example, the famous Tolstoy quote about unhappy families being different and happy families all looking alike uses schastlivyi.
Where are you getting this info from?
I am a native speaker, and the two words one would typically use for “lucky” in Russian are “везучий” or “удачливый”. I’ve never heard of“счастливый” being used to refer to any kind of luck or even allude to it. If someone asked me to translate that word, it is straight up the exact same as what “happy” is in english, no different/extra connotations or anything.
"счастливый случай" (lucky chance), guy probably meant
That doesn’t mean that “happy” (in english) is synonymous with “lucky” or that it makes sense to apply “happy” the same way outside of that saying in english. Same thing in russian with "счастливый случай". If you use "счастливый” to apply to something outside of that saying, it will mean “happy” 100% of the time. If you try using it in place of “lucky” in russian, I can guarantee that people will be very confused even after you explain your reasoning for doing so.
P.S. “случай” literally means accident/happenstance, it doesn’t mean chance. There is a word for chance in russian that is extremely commonly used, and it is pretty much the same as the english word (“шанс” pronounced as “shahns”).
Most dictionaries will offer both happy and lucky side by side (with "happy" being the first definition, "lucky" the second). See the links for some textual examples.
In other households it could very well be that you use "везучий" or "удачливый" more often, but mine (grant it, very far from Moscow) never has.
1 - https://dictionary.reverso.net/russian-english/счастливый 2 - https://dictionary.cambridge.org/dictionary/english-russian/...
There's so much hidden in homonyms, it almost feels illegal to mention it.