I want to gift an Amazon Alexa Dot but keep the Music membership on my own account - is the possible
Not unless you intend for the gift recipient to use/access your Amazon account. The device and Music subscription need to be associated with the same Amazon account. Which AMU subscription do you have? Were you planning to access music on the subscription, as well? (Only the Family Plan allows for simultaneous streaming to multiple devices.)
ETA: if you're inquiring about some kind of bundle offer (i.e. Dot includes AMU Single Device Plan), no. The Music plan is provided to the owner/user of the Dot, available once they register it to their Amazon account.