I'm a professional forum administrator, maybe I can help shed some light on the subject. Identifying sockpuppets (we call them "Alternate accounts"[1]) is not an exact science, you'll usually have to make some sort of judgement call based on the information available, the factors we consider are IP addresses, content and motive.
An IP address is like a phone number, they are unique and can only belong to one network at a time but they can be reassigned, meaning a post made from IP Address 198.51.100.0 in 2016 is not necessarily being made from the same network that made a post from that IP Address in 2015. Like a call from a phone number, a post from an IP address proves only that the post came from that IP address, it does not prove who that IP address belongs to nor does it prove who was using the network at the time, but it provides enough information for a judgement to be made.
If you receive a phone call from 07700 900000 on Monday and then you receive a phone call from 07700 900000 on Wednesday you can't know for sure that the number belongs to the same person, nor can you know who is using the phone, however you can infer based on the fact that you're being called from a number that called days before and the contents of the conversation what the likelihood is that it's the same person. If each phone call is a different voice, talking about a different subject, they might not be related, if the voice and the subject is the same: probably the same person.
Much like how the same IP address doesn't guarantee it's the same person, different IP addresses don't guarantee it's a different person. People who wish to be more secretive about their use of alternate accounts will often connect to alternate networks (usually a vpn or proxy, but maybe they go to a friends house and use their WiFi), and in cases like that it often comes down to the content (what they're talking about, the style of their posts, what motive they might have).
Regarding the technicals of IP Addresses, an IP Address is specific to a network, not device. A laptop, computer and iPad connected to the same WiFi network will all have the same public IP Address, and these devices could all belong to different people. If you were to come to my apartment and connect to my WiFi and make a post on this forum from your phone it would show the same IP address as the one this post is being made from.
[1] Sockpuppet is traditionally used to refer to someone behaving deceptively, but someone may have multiple accounts for legitimate reasons, so we call them alternate accounts (vs. their main account).
(There are some simplifications in this post, tried to keep it understandable for anyone with no technology experience.)