This is, sad to say, normal and quite standard …
… if you did what so many people do and switched content DNS service from one host to another but omitted the vitally important step of ensuring that the old host published the new domain delegation data. The way that DNS works means that its possible for people, who look up your domain names often enough, to be querying the DNS server on the old host indefinitely. There's nothing nonstandard or erroneous about this. There's no problem with caches, TTLs, routers, administrators, or any other people's DNS servers apart from your own. The DNS is working as specified. It is simply that, as specified, things can go wrong this way if you don't do switchovers correctly. (The DNSSEC people have recently discovered that DNSSEC amplifies the problem, especially if the people running the old content DNS servers choose not to coöperate.) This is why comprehensive instructions on switching DNS hosting services always point out the necessity of this step.
Of course, this is assuming that this is anything to do with the DNS in the first place. The idea that your client has two month old DNS information is sheer guesswork on your part. You haven't even checked with the client as to what DNS information xe actually has. Collect data and diagnose the problem first, before handing out spurious advice to customers to do random things to their machines.