It is quite possible.

Any bulb can produce seeds from which you can grow the same.
Just consider regular onion; If you plant one it will grow
(just like hyacinth), it will flower (like hyacinth does) and will make seeds( like YOUR hyacinth)
I have seen lots of daffodills to grow seed. Actually I have one right now that is almost dry, its head is hanging down and has a seed pod. O! Goodness, I have two of them.
Now that you asked this question, I am going to let my daffodill seed pods completely ripen and dry and I will find out how to plant them.
Some plants with rhizmus (like potato) also flower and make seeds. So there are two ways to plant potatoe;
One from seed (like tomato seeds) and the other using a potato rhitzmus.

Generally, a plants purpose of blooming is to produce seeds and by that means to propagate its gene, to guarantee the survival his specey.
In potato and bulb plants they have two alternative. Just like a lot of plants, both multiply from seeds and roots.

Let your hyacinth seeds dry naturally and experiment with planting and growing more hyacinths. I am going to try my daffodills also.

My hyacinth's and daffodils have pods on them. I'm hoping to harvest seeds and grow them as well. I love growing bulbs from seed!