What exactly does IPv4 NameServer mean? Is it the same as DNS ? Some questions regarding both

Network details and router config table

So here is my Public IP, which is shown as my router's main IP address as well.

Now I have 2 questions.

1. Gateway - In my OS network details, the gateway shows address which is used to access router configuration. But in my router's configuration, Gateway address is shown differently, does it mean my home router is connected to another router? which is owned by ISP which probably connects different routers at different user's homes.

2. DNS : In my router's config it shows 3 DNS address, the one which is marked with 1, matches my IPV4 Nameserver, so the other 2 are what for different devices connected to my home router? What exactly does that address mean?

I am aware DNS does is resolving domain names to IP address, but why is dns allocated to a machine if it already has an IP address?

What exactly does IPv4 NameServer mean? Is it the same as DNS ?

A "nameserver" is the same thing as a "DNS server".

An "IPv4 nameserver" means a DNS server that you're specifying by its IPv4 address (as opposed to IPv6 address).

(Note that any nameserver can answer any kind of query – the 'IPv4' only describes how the nameserver itself is reached, but not what it does.)

Gateway - In my OS network details, the gateway shows address which is used to access router configuration. But in my router's configuration, Gateway address is shown differently, does it mean my home router is connected to another router? which is owned by ISP which probably connects different routers at different user's homes.

Yes. The 'gateway' is "the next router" – which then has its own gateway configuration. The whole Internet is built out of routers.

(Smaller routers and other devices use just one default gateway for everything, while the larger routers used at ISPs keep a whole list of specific gateways for different networks, i.e. the "route table".)

DNS : In my router's config it shows 3 DNS address, the one which is marked with 1, matches my IPV4 Nameserver, so the other 2 are what for different devices connected to my home router?

Your computer is most likely using all three – the "network status" popup in KDE actually doesn't show more than one nameserver address. Check /etc/resolv.conf or the nmcli and resolvectl commands to see the complete list.

However, this can also depends on the router. Some routers would advertise all three nameserver addresses; some would only advertise the first one and ignore the rest; and yet others would advertise themselves as the nameserver (so that they could cache recent queries locally).

It's fairly common to provide multiple nameserver addresess for reliability – if the 1st server doesn't reply for some reason, the OS will try asking the 2nd one, then the 3rd one.

I am aware DNS does is resolving domain names to IP address, but why is dns allocated to a machine if it already has an IP address?

So that the machine would know how to resolve domain names to IP addresses, of course.