Should I keep the mysterious serum from the mission special delivery? Does it affect the story whether I keep it or give it?

No, you can get a lifetime supply later on, just keep playing.

The game itself has no consequence, but if you keep it, it may create a glitch where an NPC disappears, I would rather just give it to avoid the glitch.

Depends on which option the OP picks for the last quest in the series. One involves a lifetime supply of Mysterious Serum while the other involves a decent unique weapon. Give the Mysterious Serum since you will get a few Mysterious Serum in the next mission. The Mysterious Serum is also useful in exploring the Glowing Sea since it removes 10 rad per second for an hour.

Oh we get a lifetime supply later on? I'm just asking because I just finished the Cabot mission tonight and gave the serum.

It's one serum in inventory I think, you have to go back to re-fill it.

Correct.

There are Seven Vials available to loot in the last part of the Asylum

You could keep the first Vial You encounter, and have Eight Vials.

Siding with Lorenzo means a Lifetime Supply, one Vial at a time.

I think what's important is also the implication of having the serum itself in infinite number though for role playing purposes. Keep playing, while the gun is good, it doesn't have as much impact as having the lifetime supply of serum.

It gets rid of rads slowly like 10/s for an hour, I love using it for the glowing sea cause I hate power armor and hazmat suits.