Hi and welcome czzer. Nice birthday present!

I'm not sure how easy it would be to get that "open grain" finish on Basswood. I have achieved something similar with an Ash body. See the last picture at:
https://www.buildyourownguitar.com.a...l=1#post225886.
It was sanded, stained and then given coats of Tru-oil as the final coats. It was only lightly sanded back between coats. No grain-filler was used, no final polishing.

As a close approximation I would try:
black stain with coats of wipe on poly (satin),
or
black spray paint with satin clear coats.

I recall a forum member using a (soft!) wire brush to deepen to grain on a build, but the wood was not basswood. But I can't find the link.