That updated package seems to work fine, kernel module successfully builds and network connects.

$ uname -a
Linux athlon-desktop 6.5.0-21-generic #21~22.04.1-Ubuntu SMP PREEMPT_DYNAMIC Fri Feb  9 13:32:52 UTC 2 x86_64 x86_64 x86_64 GNU/Linux

$ lsb_release -a
No LSB modules are available.
Distributor ID: Ubuntu
Description:    Ubuntu 22.04.4 LTS
Release:        22.04
Codename:       jammy

$ apt show r8168-dkms 
Package: r8168-dkms
Version: 8.049.02-1ubuntu1.3
Priority: extra
Section: universe/kernel
Source: r8168
Origin: Ubuntu
Maintainer: Ubuntu Developers <email address hidden>
Original-Maintainer: Andreas Beckmann <email address hidden>
Bugs: https://bugs.launchpad.net/ubuntu/+filebug
Installed-Size: 1 377 kB
Depends: dkms (>= 2.1.0.0)
Homepage: https://www.realtek.com/en/component/zoo/category/network-interface-controllers-10-100-1000m-gigabit-ethernet-pci-express-software
Download-Size: 106 kB
APT-Manual-Installed: yes
APT-Sources: https://mirror.dkm.cz/ubuntu jammy-proposed/universe amd64 Packages
Description: dkms source for the r8168 network driver
 r8168 is the Linux device driver released by RealTek for their network
 controllers with PCI-Express interface:
  * 10/100/1000M Gigabit Ethernet: RTL8111B, RTL8111C, RTL8111D, RTL8111E,
    RTL8111F, RTL8111G(S), RTL8111H(S), RTL8118(A)(S), RTL8119i, RTL8111L,
    RTL8168B, RTL8168E, RTL8168H, RTL8111DP, RTL8111EP, RTL8111FP, RTL8411,
    RTL8411B
 .
 This driver should only be used for devices not yet supported by the
 in-kernel driver r8169. Please see the README.Debian for instructions how
 to report bugs against r8169 that made it necessary to use r8168-dkms.
 .
 Installation of the r8168-dkms package will disable the in-kernel r8169
 module. To re-enable r8169, the r8168-dkms package must be purged.
 .
 This package provides the dkms source code for the r8168 kernel modules.
 Kernel source or headers are required to compile these modules.

$ lsmod | grep r8168
r8168                 630784  0

$ lspci -vvv
02:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 0c)
        Subsystem: Micro-Star International Co., Ltd. [MSI] RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller
        Control: I/O+ Mem+ BusMaster+ SpecCycle- MemWINV- VGASnoop- ParErr- Stepping- SERR- FastB2B- DisINTx+
        Status: Cap+ 66MHz- UDF- FastB2B- ParErr- DEVSEL=fast >TAbort- <TAbort- <MAbort- >SERR- <PERR- INTx-
        Latency: 0, Cache Line Size: 64 bytes
        Interrupt: pin A routed to IRQ 28
        IOMMU group: 1
        Region 0: I/O ports at d000 [size=256]
        Region 2: Memory at fe100000 (64-bit, non-prefetchable) [size=4K]
        Region 4: Memory at d2100000 (64-bit, prefetchable) [size=16K]
        Capabilities: <access denied>
        Kernel driver in use: r8168
        Kernel modules: r8168

$ ip a
1: lo: <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000
    link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
    inet 127.0.0.1/8 scope host lo
       valid_lft forever preferred_lft forever
    inet6 ::1/128 scope host 
       valid_lft forever preferred_lft forever
2: enp2s0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc fq_codel state UP group default qlen 1000
    link/ether d8:cb:8a:59:f8:e5 brd ff:ff:ff:ff:ff:ff
    inet 192.168.50.104/24 brd 192.168.50.255 scope global dynamic noprefixroute enp2s0
       valid_lft 85962sec preferred_lft 85962sec
    inet6 2a00:11b1:10a0:3d20:6ed5:af4d:3483:8f66/64 scope global temporary dynamic 
       valid_lft 280sec preferred_lft 100sec
    inet6 2a00:11b1:10a0:3d20:70f2:a29e:4bf8:a9ce/64 scope global dynamic mngtmpaddr noprefixroute 
       valid_lft 280sec preferred_lft 100sec
    inet6 fe80::2bb:4112:c163:fd44/64 scope link noprefixroute 
       valid_lft forever preferred_lft forever

$ ping -c5 google.com
PING google.com(prg03s12-in-x0e.1e100.net (2a00:1450:4014:80e::200e)) 56 data bytes
64 bytes from prg03s12-in-x0e.1e100.net (2a00:1450:4014:80e::200e): icmp_seq=1 ttl=117 time=34.4 ms
64 bytes from prg03s12-in-x0e.1e100.net (2a00:1450:4014:80e::200e): icmp_seq=2 ttl=117 time=29.6 ms
64 bytes from prg03s12-in-x0e.1e100.net (2a00:1450:4014:80e::200e): icmp_seq=3 ttl=117 time=27.0 ms
64 bytes from prg03s12-in-x0e.1e100.net (2a00:1450:4014:80e::200e): icmp_seq=4 ttl=117 time=26.5 ms
64 bytes from prg03s12-in-x0e.1e100.net (2a00:1450:4014:80e::200e): icmp_seq=5 ttl=117 time=26.2 ms

--- google.com ping statistics ---
5 packets transmitted, 5 received, 0% packet loss, time 4006ms
rtt min/avg/max/mdev = 26.177/28.711/34.369/3.076 ms