If The Ownership of a Song Changes, Would I Need To Have a New ISRC Assigned?

The ISRC belongs to the recording.

No. An ISRC uniquely identifies the recording to which it was assigned, and changes in ownership should not affect a recording's ISRC.

ISRC stands for International Standard Recording Code. It is a unique code that represents the identity of your recording. In order to properly register, map, and collect royalties for a song, it is important to include an ISRC with your song registration(s).

You can learn more about ISRC's and their function here!