I have a website which has the address:
I would like to know if it can have its own DNS server like Route 53 (NS records, CNAME, A, etc.) or it is tied to the
It's a complicated answer if it's explained clearly. The domain's DNS will have to point the subdomain to the IP of the subdomain's DNS server (new DNS server explicitly for subdomain).
If you require a simple answer, yes. Absolutely. You can have another DNS server. No problems. You should explicitly specify in the DNS of the domain to point this subdomain to the new server or use the DNS that comes free with the domain.