To record 10 minutes of speech, you can expect to need 5 to 15 megabytes of storage. The greeting card IC's are not likely to have sufficient memory, nor sound quality.
A modern handheld digital recording device can do the job.
Cassette recorders are available, although less common than they used to be.
You should seek recorder modules able to plug external SD/USB memories.
This would turn your project more flexible with regard to the codec and music itself.