Is it fake? Seiko

It's a 'bull head' Seiko chronograph. I love these.

Bracelet looks fine. Enjoy!

Probably not, although it is a genuine Seiko bracelet and may have been a replacement for a damaged original.
It doesn't really matter, you have a very nice Bullhead.

The bracelet is period correct for your watch. It may or may not be original. It's not worth the time trying to find out. If you like it, wear it. If you don't like it, swap it out. Not much value placed on Seiko bracelets as there are billions out there.
If your Bullhead runs and keeps time, wear the heck out of it. You will get plenty of compliments. Not sure what it costs for a full service.

Seiko bullhead 6138-0040 came in 2 flavors, the brown “Root beer” and the “Black Horse” with black dial and blue subdials.
It’s a great movement and highly collectible watch. It is very large but fun to wear. Here’s a good overview:

https://www.fratellowatches.com/seiko-watches/tbt-seiko-bullhead-6138-0040/#gref

It came on a fish bone bracelet seen on my black one, and an oyster bracelet on the Root beer pictured. I agree with others that Seiko bracelet you have is fine I wouldn’t change it. It is heirloom might as well wear it as your father did. If you like it and plan to wear it I would find a vintage Seiko friendly watchmaker and have it serviced They are great watches and fun to wear.