Yes, D. Attenborough probably refers to ploidy number
Humans for example are 2N (except during the spermatozoid and ovule phase of human existence) meaning that they carry two copies of each autosome (=non-sexual chromosomes). Some species are 1N (haploid), some are 3N (triploid), etc… It would actually be more correct to talk about the time a species spend in each of its phase or talk about the number of cells in each of its phase because many species undergo change in chromosome number during their existence as it is the case for humans. Humans are 2N most of the time but they are 1N when they are ovules or spermatozoid. The correct word to describe the human lifecycle is diplontic. Some species are diplontic, some are haplodiplontic and some have phases when they have a greater number of chromosomes. Those species are said to be polyploid.
Ploidy number, chromosomes and chromatids
To make sure one don't mix up things... One chromosome is made of one or two chromatids (depending on the cellular phase). The chromatids of the same chromosome (called sister chromatids) are (almost completely) identical. A given "chromosome type" is either alone (1N) or accompagnied by one very similar chromosome (2N) or two other very similar chromsomes (3N), etc.. Considering the case of a 2N individual: If at a given locus (position in the chromosome) the two alleles (gene variant) are different than the individual is said to be heterozygote for this locus. The ploidy number describe "this number of chromosomes"! Now there may have different number of different "types of chromosomes". Humans for example have 23 different types of chromosomes and in its haploid phase has therefore 23 chromosomes (23 or 46 chromatids depending on the cellular phase) and has 46 chromosomes (46 or 92 chromatids depending on the cellular phase) in its diploid phase
Ploidy number in angiosperms and other plants and non-plants organisms
Yes, D. Attenborough (who I don't know and I haven't read anything of his work) probably means doubling, tripling, quadrupling, .. the number of N of chromosomes. And, yes indeed polyploidy (3N, 4N, 6N, 8N) is relatively often observed in flowering plants. There are many crop species that are polyploid. According to this article an important part of the speciation that occurred in plants is accompanied by a change in ploidy number but this is not only true for angiosperms.
Polyploidy is also found in many fungi species and some animal species as well. There are lots of terminologies linked with the discussion of chromosome number and reproductive system and many articles discuss these mechanisms and also the evolution of polyploidy. I found the work of Sally Otto to be of the greatest interest for example.
The words inventions are not really used in evolutionary biology but rather in popular books. Also, there are many non-angiosperm plants that have symbiotic relationship with animals. For examples: some angiosperms and the squirrels or birds. But it is true that angiosperm have particularly important symbiotic relationship with animals, in particular insects and this probably allowed an important diversification in both angiosperms and insects. As already said angiosperms are not the taxon of plants to include polyploid species. There are other characteristics that allow to tell an angiosperm apart from other plants. The number of aperture on pollen grain or the positions of the meristems (tissue where mitotic division of unspecialized cells occur allowing the growth of the plant) for examples.
You may be interested in reading this post on the effect on fitness of ploidy number