In my game, I'm an elvish druid who was gifted a Blink Dog familiar, and I am trying to find the familiar's owner (a wizard). The dog follows me around, but out of my sight. I have attempted to call it the way my master has (using its favorite toy), but it has yet to try and collect it like it had before.
How else should I call upon this creature? Is it possible to begin to train it to follow my lead instead of its original owner?
A blink dog is a fey, not a beast. Furthermore, they understand Sylvan language and are as intelligent, wise and charismatic as the average humanoid, if not more.
You are approaching this from the wrong mindset: a blink dog is a person - you can befriend one, even educate one but you can't train one like it is an animal.
The specifics of this situation are really going to be driven by your DM. We can't tell you what to do because this sounds like it's part of the story being told by your table. How you can interact and what the goals of the DM are with the blink dog are really up to them.
I'd recommend just to play as you would, talk to your DM, and see where the story goes.
I don't know the answer to this as the rules don't cover this. When the rules don't cover something, it is up to the DM to determine. Ruling either way is acceptable (after all, humans are animals and we're clearly trainable) and it will come down to how the table and the DM wants to treat it.
Maybe trying to train it is part of the story your DM is telling and this is a character growth opportunity. Who knows? Your DM.
But, either way, you need to establish trust with this fey. The blink dog may have been attracted to the toy from its master because it was it's favorite toy AND because it trusted its master.
You've only got half the equation: the toy. Now you need to work on building trust and a relationship.
A blink dog is smart - having the same intelligence, on average, as the average human. They can also understand a language (sylvan) and can speak a language known as "Blink Dog".
So if you can train a person you can train a blink dog. But they are not pets - if you want them to do stuff, you will need more than a snack or the promise of walkies - you will need to negotiate.
Try talking to it as an equal rather than treating it as a doggy. Ask what it would accept as payment in return for its services.
If you need to have a proper conversation you can use magic, such as the third level divination spell Tongues.
