The bubbles are air, not water
When water pours, it does not fall as a solid plug. The moving stream pulls air along with it, and that air gets pushed below the surface, where it shows up as bubbles. This is why the effect depends on how the water is poured rather than on the water alone: the geometry of the pouring setup changes how much air is entrained [2].
The same trapping happens when a jet of grains is poured into water, which also drags air down and generates bubbles [3]. That is a useful comparison because it shows the bubbles are not some special product of water itself. The common ingredient is a falling material that carries air into the liquid.
Why some bubbles cling to the container wall
Bubbles form in all kinds of containers, but containers made of d-block metals such as copper and steel show a distinct behavior: the bubbles stick to the walls after forming and stay mechanically and structurally stable [1]. One proposed explanation is that hydrogen sitting in the spaces between atoms of these metals forms hydrogen bonds with the water bubbles, which could account for that stability [1]. This is a hypothesis the authors tested rather than a settled mechanism, so it is best read as a possible reason the bubbles hold on.
The practical consequence is that the container is part of the story. Two pours that look identical can leave different amounts of clinging bubbles depending on what the container is made of [1].
What changes how many bubbles you get
The number of bubbles is not fixed. In experiments that poured water from different heights, into containers of different cross-sectional areas and materials, and at different water temperatures, the researchers derived mathematical relations for predicting bubble numbers from those starting conditions [1]. In plain terms, pour height, container size and shape, material, and temperature all shift the result.
The pouring setup itself is a lever too: the design of the pouring basin and the channel the water flows through affects how much air is entrained during pouring [2]. So if you want fewer bubbles, changing how the water is delivered is as relevant as changing the water.
Not every bubble in water comes from pouring
Bubbles also form in water for reasons unrelated to pouring, and it helps to keep those separate. During freezing, bubbles nucleate and grow at the boundary between ice and water, with the initial air concentration and the freezing rate shaping the outcome [4]. In porous electrodes used for alkaline water electrolysis, gas bubbles are generated by the electrochemical reaction, and the pore structure determines whether bubbles get trapped or escape; tortuous pores tend to accumulate bubbles, while straight pores seldom trap them and allow merged bubbles to jump away from the wall [5].
Those cases involve different causes, so they should not be used to explain a bubble you see while pouring a glass of water. The pouring explanation is about air carried into the liquid by the falling stream and, in some containers, held against the wall [1][2].
Sources used in this answer
Formation and sticking of air bubbles in water in d-block containers
In d-block metal containers such as copper and steel, bubbles formed by pouring water stick to the walls and are mechanically and structurally stable, with interstitial hydrogen proposed as a possible cause, and bubble numbers can be predicted from pour height, container cross-section, material, and water temperature.
Water modelling of effects of pouring basin and sprue geometry on entrance of air bubbles into mould
The design of the pouring basin and sprue affects how much air is entrained as bubbles during pouring, studied through water modelling.
Air entrainment and granular bubbles generated by a jet of grains entering water
Pouring a jet of grains into water entrains air and generates bubbles, analyzed with high-speed imaging across varied mass, grain size, temperature, and impact height.
Nucleation and growth of bubbles at an ice–water interface
Bubbles nucleate and grow at a growing ice-water interface during freezing, governed by initial air concentration and the water conditions.
Bubbles in Porous Electrodes for Alkaline Water Electrolysis
In porous electrodes for alkaline water electrolysis, tortuous pores trap gas bubbles while straight pores seldom do, and merged bubbles near walls can jump away.