If two computers connect to the same domain name, do they share an internet connection and network?

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Your question is worded sort of awkwardly. The answer could be yes or no, but probably yes.

What are you doing to see that they have the same domain name? If you're outside your network, and you're tracing back to your IP address, and you see something like DSL-10.10.10.45.subscribers.att.net as the domain name, then yes, they share an internet connection. On your average home internet connection, you get one IP, and one DNS name, and they are assigned to your DSL or cable modem, not your computers. Everything behind the modem appears to have the same DNS and IP to the outside world.

You also ask if they are on the same network connection as opposed to the same internet connection. This is not necessarily true. If I have two routers in my house, and they both connect to my cable modem, I could have computers connected to both routers using different private addressing, and they would still appear to have the DNS and IP of the modem to the outside world. However, how many homes have more than one router and network segment? Not too many.

Is this a home or business connection? Do you mean the subdomain as well as the second-level domain? Do they also share the same public IP address when viewed from the outside?

Added:
If they have very different public IP addresses, they are very likely on different internet circuits. An internet connection to a home or small business is usually assigned one IP or a small range of IPs, like 64.64.64.1-64.64.64.4. If the masks and IPs are not contiguous addresses, I would say they are on different internet connections. I would really need to see more to say why they share domain names. Again, I'm wondering if you're talking about:

pc1.myinternet.com
and
laptop1.myinternet.com

or, that a DNS lookup of both devices shows

computer.myinternet.com

One possible explanation is that it is internal, not external domain names in use in one or both cases.

However, they would not function properly on the same network segment if they had different IPs but were both called computer.myinternet.com. DNS would not resolve to an IP properly.

Hope that helps!