In English you might say “my Spanish is not very good,” in reference to your language skills, or “I’m trying to improve my Japanese.” Is the same true in French?
Yes, we definitely use the same phrasing.
Beware that there is no capital in language names in French:
Mon espagnol n'est pas très bon.
J'essaie d'améliorer mon japonais.
Yes, you can say it, but this is, in my opinion, not the best option.
I personally never heard "mon" before a language! I would use "en":
Je ne suis pas très bon en espagnol
J'essaye de m'améliorer en japonais
Literally, your language is not improving, but you are improving in language.