How can I check if I'm connected to my own wireless network on windows or linux?

Check the BSSID, that is, the access point's MAC address.

At home you'll usually have only one access point that's built into your router, so compare the MAC with the one written on the sticker on its bottom. (It might differ by a single digit in some cases.)

You didn't mention the OS you use, so I'll assume Linux:

$ iw wlan0 link
Connected to 24:a4:3c:ce:d2:16 (on wlan0)
    SSID: eduroam
    freq: 2462
    ...

Or you could check whether a scan detects multiple APs with the same name:

$ sudo iw wlan0 scan | egrep "^BSS|SSID"

$ nmcli -f in-use,ssid,chan,bars,security,bssid dev wifi list

There are various "Wi-Fi scanner" apps for Windows and macOS as well – can't recommend any, but most of them are able to display a basic "network list" including channels, BSSIDs, and all that. There's one from Nirsoft, and probably heaps of other freeware.

If you're on Windows (7 or later), the same can be done via netsh:

C:\> netsh wlan show interface
There is 1 interface on the system:
    ...
    SSID                   : eduroam
    BSSID                  : 24:a4:3c:ce:d2:16
    Network type           : Infrastructure
    Radio type             : 802.11n
    ...

(I can't remember offhand what's the command to request a scan, but it's also under netsh wlan show <something>.)

On macOS, this post suggests:

/System/Library/PrivateFrameworks/Apple80211.framework/Versions/A/Resources/airport -I en1
              ...
      link auth: wpa2-psk
          BSSID: 24:a4:3c:ce:d2:16
           SSID: eduroam
            MCS: -1        channel: 6

On FreeBSD, all that should be under ifconfig.

Android has quite a few "Wi-Fi analyzer" apps; at the time of posting, this one worked well.

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The homepage, however, is not a network setting. Only your web browser cares about that.

If you have a Linux machine (I assume you can do this on Windows too), you can run arp <gateway address> to get the MAC address of the router (labelled HWAddress). Compare this to the MAC address that is printed on the sticker on the router (or at least, should be). E.g.

[root@server~]# arp 192.168.1.1
Address                  HWtype  HWaddress           Flags Mask            Iface
gateway.domain.com ether   XX:XX:XX:XX:XX:XX   C                     eth0

[EDIT]the Windows syntax is:

arp -a <ip address>