One of my ancestors has this on death certificate
In the Lair of 'The late Robert Aitken'
What does this mean precisely and would their be any other record?
First, it tells me that whoever it was died between 1855 and 1860, because before 1855 there was no such thing as a death certificate, and after 1860 the place of burial is not included in the death certificate.
When cemeteries as opposed to kirkyards began to be provided, people would buy a lair. A lair is a grave large enough for one lot of coffins to be buried on top of one another. Sometimes people would buy two or even three lairs together, so their grave would have space for two or three lots of coffins.
So this means that Robert Aitken had bought the right for his family to be buried in a particular lair, and this person was buried in that lair. So it is reasonable to suppose that there was some connection between them.
Also, it means that Robert Aitken was already deceased before this person.
There would normally have been records of burials, which would list the names and dates of burial of everyone buried in the cemetery. I would expect such records, if they have survived, to be in the care of the City of Glasgow council. Start by looking in the catalogue at Glasgow City Archives
https://www.glasgowlife.org.uk/libraries/city-archives.
I have in my tree a Robert Aitken, builder, who lived in Glasgow and died on 31 August 1827. He was married twice; first to Agnes Clark and second to Jean Shanks. There were eight children of the first marriage, including a Robert, and six of the second marriage.