How to enable ethernet 1 Gigabit advertising or troubleshoot the situation?

On an Debian Stretch box, I have a Gigabit ethernet network adapter, but this mode is not active (however displayed as supported).

Does this depend on the peripheral condition (i.e. cable is not good enough, another one box has an older i.e. only 100 Mbit network controller - or can I somehow activate this mode?

Output of ethtool eth0:

Settings for eth0:
        Supported ports: [ TP MII ]
        Supported link modes:   10baseT/Half 10baseT/Full
                                100baseT/Half 100baseT/Full
                                1000baseT/Half 1000baseT/Full
        Supported pause frame use: No
        Supports auto-negotiation: Yes
        Advertised link modes:  10baseT/Half 10baseT/Full
                                100baseT/Half 100baseT/Full
        Advertised pause frame use: Symmetric Receive-only
        Advertised auto-negotiation: Yes
        Link partner advertised link modes:  10baseT/Half 10baseT/Full
                                             100baseT/Half 100baseT/Full
        Link partner advertised pause frame use: Symmetric
        Link partner advertised auto-negotiation: Yes
        Speed: 100Mb/s
        Duplex: Full
        Port: MII
        PHYAD: 0
        Transceiver: internal
        Auto-negotiation: on
        Supports Wake-on: pumbg
        Wake-on: g
        Current message level: 0x00000033 (51)
                               drv probe ifdown ifup
        Link detected: yes

The key lines are:

    Link partner advertised link modes:  10baseT/Half 10baseT/Full
                                         100baseT/Half 100baseT/Full

This likely means that the other device (most likely a switch) is not advertising 1Gbps, or that the cable in question isn't capable of supporting 1Gbps. Check both, and verify that the cable has all eight pins connected - if in doubt, switch out the cable.

You can also use mii-tool -v to verify the information from ethtool.

Alternatively, you can use ethtool to force the NIC to 1Gbps, but be aware you may lose network connectivity:

ethtool -S eth0 speed 1000 duplex full autoneg on