Why do some words with similar meanings sound similar as well?

I just noticed while writing a few examples of similar words that uncannily sound like each other phonetically.

Examples: An example is the similar words: “gleaming”, “glittering”, “glinting”, and “glimmering”.

Also: “repulsive” and “repugnant” are interesting.

There are more words like this but I can’t think of them immediately on the spot. Obviously English is a rich language and thus one can find numerous examples seemingly showing different things, but is this to do with the roots of the words? As in, do they actually share the same etymology or are they just coincidental?

Yes, all of your examples share the same etymology and can be traced back to the PIE root *ghel. As explained in etymonline.com (emphasis mine):

*ghel- (2)

Proto-Indo-European root meaning "to shine;" it forms words for "gold" (the "bright" metal), words denoting colors, especially "yellow" and "green," also "bile, gall," for is color, and a large group of Germanic gl- words having to do with shining and glittering and, perhaps, sliding. Buck says the interchange of words for yellow and green is "perhaps because they were applied to vegetation like grass, cereals, etc., which changed from green to yellow."

It forms all or part of: arsenic; Chloe; chloral; chloride; chlorinate; chlorine; chloro-; chloroform; chlorophyll; chloroplast; cholecyst; choler; cholera; choleric; cholesterol; cholinergic; Cloris; gall (n.1) "bile, liver secretion;" gild; glad; glance; glare; glass; glaze; glazier; gleam; glee; glib; glide; glimmer; glimpse; glint; glissade; glisten; glister; glitch; glitter; glitzy; gloaming; gloat; gloss (n.1) "glistening smoothness, luster;" glow; glower; gold; guilder; jaundice; melancholic; melancholy; yellow; zloty.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit harih "yellow, tawny yellow," hiranyam "gold;" Avestan zari "yellow;" Old Persian daraniya-, Avestan zaranya- "gold;" Greek khlōros "greenish-yellow color," kholos "bile, gall, wrath;" Latin helvus "yellowish, bay," Gallo-Latin gilvus "light bay;" Lithuanian geltonas "yellow;" Old Church Slavonic zlutu, Polish żółty, Russian zeltyj "yellow;" Latin galbus "greenish-yellow," fellis "bile, gall;" Lithuanian žalias "green," želvas "greenish," tulžis "bile;" Old Church Slavonic zelenu, Polish zielony, Russian zelenyj "green;" Old Irish glass, Welsh and Breton glas "green," also "gray, blue;" Old English galla "gall, bile," geolu, geolwe, German gelb, Old Norse gulr "yellow;" Old Church Slavonic zlato, Russian zoloto, Old English gold, Gothic gulþ "gold;" Old English glæs "glass; a glass vessel."

The same principle holds for repugnant and repulsive both of which are derived from Latin words using the word forming element -re meaning "back to the original place; again, anew, once more". In the case of repugnant, the etymology (according to etymonline) is (emphasis mine):

repugnant (adj.)

late 14c., "contrary, contradictory," from Old French repugnant "contradictory, opposing" or directly from Latin repugnantem (nominative repugnans), present participle of repugnare "to resist, fight back, oppose; disagree, be incompatible," from re- "back" (see re-) + pugnare "to fight" (from PIE root *peuk- "to prick"). Meaning "distasteful, objectionable" is from 1777.

And that of repulsive is:

repulsive (adj.)

early 15c., "able to repel," from Middle French repulsif (14c.) and directly from Medieval Latin repulsivus, from repuls-, past participle stem of repellere (see repel). The sense of "causing disgust" is first recorded 1816. Related: Repulsively; repulsiveness.

While repel's is (emphasis mine):

repel (v.)

early 15c., "to drive away, remove," from Old French repeller or directly from Latin repellere "to drive back," from re- "back" (see re-) + pellere "to drive, strike" (from PIE root *pel- (5) "to thrust, strike, drive"). Meaning "to affect (a person) with distaste or aversion" is from 1817. Related: Repelled; repelling.

As you can see, both words can be traced back to the same -re.

So yes, for both the sets you mention in your question, the reason they sound similar is that they share similar etymologies. I believe it is this shared origin which gives rise to the phonaesthetic effect described in Decapitated Soul's answer. The reason these words all sound similar is because of their shared etymology, and the result of this similarity is the phonaesthetic effect.

Gleaming, glittering, glinting, glimmering, glisten etc., sound similar and have closely related meanings because they share the same phonaestheme gl-.

Repugnant, repulsive, reprehend, repent, reprove etc., can also be said to have a common phonaestheme rep-.

Sound symbolism

A phonaestheme is a particular sound sequence that suggests a certain meaning. Words beginning with gl- are often associated with light reflection or vision: glitter, glisten, glow, gleam, glare, glint, glimmer, gloss etc., pertain to light. The study of phonesthemes is called phonaesthetics and the phenomena is called phonesthesia or sound symbolism.

David Crystal has explained it in Detail in The Ugliest Words. Here's an example from The Ugliest Words [modified & simplified]: Suppose you're in a spaceship approaching a different planet where there are two races. One of them is beautiful and friendly, the other is unfriendly and ugly. You also know that one of them is called Lamonians and the other is called Grataks. Which is which? David Crystal says that most people associate Lamonians with the friendly race and Grataks with the bad guys. He further says that it's all a matter of sound symbolism and that words having soft sounds (nasals & liquids, long vowels/diphthongs etc), reinforced by a gentle polysyllabic rhythm, are interpreted as "nicer" than words having harsh sounds (such as the velar plosives), short vowels and an abrupt rhythm.

Examples

There are many other examples of phonaesthemes. Some examples are:

• Sl-

words starting with sl- are often associated with pejorative behaviours, traits, or events: slack, slouch, sludge, slime, slosh, slash, sloppy, slug, sluggard, slattern, slut, slang, sly, slither, slow, sloth, sleepy, sleet, slip, slipshod, slope, slit, slay, sleek, slant, slovenly, slab, slap, slough, slum, slump, slobber, slaver, slur, slog, slate etc. [Wikipedia]

• Sn-

words starting with sn- are related to mouth and nose: snarl, sniff, sniffle, snuf, snoot, snout, snub, snot, sneeze and snore etc. [ThoughtCo]

• Fl-

these words are expressive of movement as in: flap, flare, flee, flick, flicker, flounder, fling, flip, flit, flitter, flow, flutter, fly, flurry, flounce, flourish, flout, flail, flash, flex, flinch, flock, flop etc. [ThoughtCo]

• Pr-

related to human (social role) as in: proud, prior, prophet, profit, proxy, prosper, pride, price, private, prize, precious, prelate, prep, praetor, prince, privy, priest, pro, pray, pram, prance, prattle, proffer, prom, prude, prompt, proper, prayer, prate, praise, prig, prim, primp, prink, prissy, pretty, preach, preen promise etc. [John Lawler's research papers] (I highly recommend reading it.)

• Squ-

(/skw-/) denotes discordant, eruptive sounds: squeal, squeak, squash (quash), squall (scream discordantly) {variant of squeal}, squish {variant of squash}, squidge {variant of squish}, squitter, squirt, squawk (cry with a harsh note), squirk (half-suppressed laugh, squeak), squank etc. [English Phonaesthemes Dictionary]

• -irl/-url

related with circular as in: twirl, curl, furl, burl, knurl, whirl, hurl, swirl, purl etc. [English Phonaesthemes Dictionary]

• Cl-

denote sound: cluck, click, clap, clack, clash, clutter, clang, clank, clamber, clamour, clam, clump, clip etc. [English Phonaesthemes Dictionary]

• G- (/g/)

associated with throat: gulch (swallow, devour greedily), gulp, gush, gaggle, giggle, gabble (jabber), gobble (said of turkey cock), gurgle, guggle, guttle (eat greedily), guzzle (swallow liquor greedily), gargle, gulch (ravine, cleft), gab (talk), gob (talk), gob (mouth, beak), guff (puff, whiff), guffaw etc. [English Phonaesthemes Dictionary]

• Y-

words starting with y can be associated with shouting, yelling, crying, chattering or other similar sounds as in: yell, yelp, yap, yatter, yammer, yay, yowl, yesk, yawl, yip etc. [English Phonaesthemes Dictionary]

• Ab-

ab- can also be associated with negative meanings: abhorrent, abhor, abrasive, absurd, abuse, abominable, abolish, abduct, aberrant etc. [Source: myself]

References

• Sound symbolism - Wikipedia
• Phonaesthetics - Wikipedia
• Phonaestheme - Wikipedia
• John Lawler's research papers
• English Phonaesthemes Dictionary
• Phonaestheme - ThoughtCo

The following papers might also be of interest:

• The Reality of English Phonaesthemes
• Sound Symbolism – Berkeley UC