Blindsight (monster ability)

A creature with blindsight can perceive its surroundings without relying on sight, within a specific radius. Creatures without eyes, such as oozes, and creatures with echolocation or heightened senses, such as bats and true dragons, have this sense.

This isn't super helpful mechanically, however it does mention 'creatures with echolocation like bats'. Echolocation in real life still requires a line of effect, and wouldn't work through solid walls. This isn't specifically spelled out in the text, but in the absence of a rule we have to take into account the words being used. In this case, a bat would be able to echolocate in complete darkness, but would not be able to echolocate a person on the other side of a 5 foot stone wall. Since the manual doesn't go on to describe echolocation in any greater detail, we have to assume it means in DnD what it means in real life.

It looks like a Purple Worm has both blindsight and tremorsense, which would seem to show that there is a reason to have both, which would again point to the above definition where blindsight does not work through solid objects but tremorsense does.

Tremorsense does not provide "sight"

To begin the description of tremorsense does not imply that it allows it to replace sight (DMG, 9):

A monster with tremorsense can detect and pinpoint the origin of vibrations within a specific radius, provided that the monster and the source of the vibrations are in contact with the same ground or substance.

This description is specific in what it allows tremorsense to detect: "the origin of vibrations". In practical terms it is easy to imagine a monster using vibrations to detect a creature's location. It is harder to imagine vibrations equalling sight for the purpose of determining where an archer is aiming or other minute details.
There is also nothing indicating that tremorsense allows a monster to pinpoint non-vibrating objects. It can hardly qualify as "seeing" if the creature cannot detect a door or a pillar in its way.

The description of tremorsense can be compared to the description of blindsight which offers more generic perception (DMG, 8):

A monster with blindsight can perceive its surroundings without relying on sight, within a specific radius.
... If a monster is naturally blind, it has a parenthetical note to this effect, indicating that the radius of its blindsight defines the maximum range of its perception.

The understanding that tremorsense, unlike blindsight, cannot replace regular vision is further supported in the passage describing Vision and Light¹. This passage explicitly mentions blindsight, darkvision, and truesight as exceptions to the regular rules for sight. It makes no mention, however, of tremorsense being an exception to the rules of sight.

If tremorsense were intended to provide sight it would say so and would be more aptly names tremorsight, or tremorvision in line with blindsight, truesight, and darkvision.

Tremorsense is not used as a substitute for sight in published monsters

Unlike with blindsight there are no monsters that rely exclusively on tremorsense for perceiving their surroundings. This makes sense in that every monster with tremorsense in the Monster Manual, except for Galeb Duhr, also has a burrowing speed. The description of tremorsense also notes that it is generally possessed by burrowing creatures. Tremorsense is then used to identify when other creatures are nearby underground. For this purpose the location of the creature is sufficient because if the monster were in a situation where sight were relevant it would be near enough to rely on darkvison or blindsight.

Excluding the Borborygmos, Mantrap, Sacred Stone Monk, and Stonemelder all published monsters with tremorsense at this time also have blindsight or darkvision with which to see. The Borborygmos has the feature Poor Depth Perception which causes disadvantage on targets more than 30ft away. If its 60ft tremorsense could replace sight then one would imagine Poor Depth Perception would only apply outside of a 60ft radius rather than outside a 30ft radius.

Of monsters with tremorsense only the Core Spawn Crawler lacks regular vision. The Core Spawn Crawler provides additional evidence that tremorsense does not allow sight because its blindsight reads:

blindsight 30 ft. (blind beyond this radius)

If the crawler's 60ft tremorsense could allow it to see then it would be inaccurate to say that it was blind beyond 30ft. The "blind beyond this radius" note would be better placed after tremorsense if tremorsense could replace sight.

How is tremorsense treated conventionally?

I cannot speak to the "average" dnd gameplay but one of the strengths of DnD-5e is its simplicity and universality. If, in fact, many players use a different understanding of tremorsense then the prevailing norm could be more important than a strict RAW interpretation. Ryan C. Thompson provides a reason in their answer to a related question to allow tremorsense to replace sight:

Ultimately, I can think of reasonable arguments for both rulings, so I'm not going to recommend one over the other. However, I believe that ruling 1² is much more common in actual play, not least because it makes monsters with tremorsense more of a threat and makes them easier for the DM to manage in battle (one less source of advantage/disadvantage to keep track of).

Concerns of simplicity are important and the convention of a table trump RAW. If a DM wants tremorsense to replicate sight that is their purview regardless of whether it derives from a different interpretation of the text or from a desire for simplicity.

^{1: Credit to Mark Wells for this argument
2: Tremorsense is effectively a substitute for vision for targets that it can detect.}