This was actually possible and worked in older patches, but was broke in some 1.8 snapshot I'm not sure when
Steps to Reproduce:
1. Create a record with a custom name:
summon Item ~ ~1 ~ {Item:{id:"minecraft:record_13",Count:1b,tag:{display:{Name:"CUSTOM NAME"}},Damage:0s}}
2. Place in record player
3. Eject it, it no longer has its custom name
Checking the record player's NBT, it actually loses the custom name when placed in the player.
Confirmed.