Most of the lexical resemblances betwen IE and Uralic look like loanwords from PIE into Proto-Uralic. The sound correspondences are such that the Uralic words have those sounds that are closest to the IE sounds, i.e. the expected sound substitutions. This is particularly striking with the vowels, where such things as ablaut grades and laryngeal colourings are faithfully reflected in the Uralic word forms. And as ablaut and laryngeal colouring are innovations within PIE that do not predate Proto-Indo-Uralic, this hits the nail home: these words just cannot be inherited from Proto-Indo-Uralic!
Yet, the morphological resemblances are hard to explain by anything other than a common ancestor. And there are a few lexical resemblances that do not show the sound correspondences expected from loanwords, and these may be genuine cognates. However, the frequently cited 'name' word faces the difficulty that PIE *hnehmn almost certainly isn't a single root, rather a derivative from an (otherwise unknown) root *hneh-, perhaps 'to call', with the well-known suffix *-m(e)n, which makes comparison with PU nimi difficult. This may be a chance resemblance, like Japanese namae. (The initial laryngeal is uncertain; it is often reconstructed as *h, but that is rather beside the point.)
This seems to be a bit like the Armenian case, where the large number of lexical resemblances to Iranian languages, on the ground of which Armenian was classified as Iranian by early Indo-Europeanists, turned out to be loanwords, but it also turned out that some other - fewer! - words that didn't fit Iranian, and especially the morphology, are cognate to those of other IE languages by a different set of sound correspondences, and Armenian is now classified as a separate branch of IE on the ground of this.
...brought to you by the Weeping Elf
Tha cvastam émi cvastam santham amal phelsa. -- Friedrich Schiller
ESTAR-3SG:P human-OBJ only human-OBJ true-OBJ REL-LOC play-3SG:A