I don't understand what they mean by mainframes were stand alone.
You left out the a very important restriction of that claim:
"In the mid-1960s, mainframe computers in research organizations were stand-alone devices."
Universities had usually rather small and limited installations, and were usually rather cash strapped. This is in contrast to commercial settings where networking was already a thing right from the start of electronic computing. Also helpful to keep in mind that networking like today wasn't really something in need. People were still exploring the idea of a computer at all.
Could they be connected to each other if they were from the same manufacturer?
Yes, of course. Remote Job Entry (RJE) is a core function of mainframe systems and dates back to punch-card times - way before terminals became a thing. RJE means subsidiaries like a local bank branch would punch their transactions, pack them into one or more jobs, add jobs to get printouts and so on. The central mainframe would fetch those jobs by operating the local reader, execute them and punch out result at local card punches or printers.
With (comparably) cheap computers becoming available during the 1960s, those branches got their own computers holding data for local display and processing.
This was pretty standard in 1960. Larger companies, especial banks, employed multi-tier structures with local computing centers doing processing for branches and exchanging data between them and toward a headquarters installation.
While manufacturers like IBM did of course prefer to sell their own products for all elements, RJE became an area where third party equipment had a foothold. In fact, third party RJE development is the very heritage of today's x86 world. x86 is a direct offspring of the Datapoint 2200 CPU, which was developed to replace classic punch-card RJE by a screen-based system with tightly integrated local data processing/handling capabilities, still using the same communication infrastructure.
By the early 1970s, vast private networks built around manufacturer protocols (mostly IBM) with multi-tier processing did exist in the US and worldwide.
Why go through all that to help scientists share their findings when they can just buy mainframes from the same manufacturer and connect them?
Money? After all, universities were at all times less well funded than banks, insurance companies or utilities. The usage of rather low-end systems for nodes and hosts does tell the same story.
Next it's a different challenge to combine vastly different systems versus integrating into a single system of a major manufacturer.
Last but not least, buying stuff does not create knowledge or educate; building your own stuff is great research work - exactly what universities are about: research and learning.
Side note:
Contrary to the common assumption, those connections weren't slow per se. Sure, dial-up lines for very small installation might have been as slow as 300 or 600 Bd, but network connections between remote mainframes were already ca. 1970 operating at 4800 or 9600 bd line speed.
Local connection between mainframes could be made even faster, either using communication processors which could deliver up to several hundred kilobytes between multiple machines on the same campus, or by coupling two mainframes using a channel to channel adaptor delivering transfer speed way past a megabyte per second. Of course the later were restricted to a distance of 200 ft between the CPUs.
