Pager technology has always allowed arbitrary alphanumeric (A-Z, a-z, 0-9, some special characters) messages of various lengths. How they are used depends on the person or company using them. Typically, short medical codes can be used, to indicate a longer or complicated situation, but proper length messages can be sent as well. Pagers can scroll messages that are longer than the display, so they are not limited to 10 or 16 or n Characters. Depending on the service really, as some have a 240 character limit. Text messages, aka SMS, in comparison are limited to 160 ASCII characters.
Tangent: No different than SMS, except most pager companies have dedicated hardware/antenna, and for medical contracts, have some minimum guaranteed service contracts. These service contracts dictate downtime, message delivery times, etc.
SMS is a fairly faulty protocol and has somewhere like 10% failure rate, requiring 15~20% resend rate. SMS is an (ab)use of the cellular side bands, bands that are normally used for control between a cell phone and tower. SMS is a giant cluster hack of a technology. SMS/Cellular technology, mainly the wavelength/bandwidth being used, was previously very bad at wall penetration, while Pager tech had much better coverage. Distance was also a factor (Cellular cell sites are often >20 miles at best, Pager sites are 100+ miles). There is also the usage of Simulcasting/broadcasting of Pager messages among multiple cells, while Cellular technology will only connect on a single cell tower. There are multiple technological reasons for Pager technology still being persuasive.
As we never see the messages on House, it's speculation on whether it is code or text.