Why is the Horklump classified as a beast?

As you see in the description of the Horklump, it is a mobile creature.

The Horklump can cover an average garden in a matter of days.

It is a common mistake that people think it grows to cover the garden, but according to Pottermore, it infests the garden, it does not grow to cover the garden.

Therefore it is a mobile creature. The definition of plant is,

A living organism of the kind exemplified by trees, shrubs, herbs, grasses, ferns, and mosses, typically growing in a permanent site, absorbing water and inorganic substances through its roots, and synthesizing nutrients in its leaves by photosynthesis using the green pigment chlorophyll.

Although this is a "Muggle" definition, it carries over to the wizarding world. The Venomous Tentacula is stationary, and so is the Mandrake. Although they do move, their roots stay in the same place. Therefore, the Horklump is then classified as a beast by it's ability to move to different locations, where other wizarding plants are stationary. Also, as this beast searches for food, it leads us to believe that it does not go through photosynthesis, and that it's main source of food are the worms that it finds.

It spreads it's sinewy tentacles rather than roots into the ground to search for it's favorite food of earthworms.

Horklumps seem to be very similar to octopi when you look at what's beneath the soil:

Fantastic Beasts and Where to Find Them: Illustrated Edition via the HP Wiki

Thus, I believe that they are considered beasts for the same reason a potted octopus is: its cells do not have rigid cell walls, like plants and fungi do (including the Venus fly trap).

Categorization is hard, even when we're dealing with the Muggle world.

... as just one not-entirely-random example, this very site was originally going be called just "Science Fiction" and then this happened. :-)

(See also How An Algorithm Feels From Inside which tries to explain why we are naturally inclined to believe that "is it a plant or not" should have a single, unambiguous, and clear-cut answer.)

That said, the Horklump does not appear to be very plant-like. Let's go through some typical characteristics that usually distinguish plants from animals, as described, e.g., here, as well as whatever other evidence we have.

• Description as a "beast". Inconclusive, because even Muggle plants are sometimes described as beasts, but if the same text doesn't describe Venomous Tentacula or Mandrakes as beasts, then this is weak evidence against the Horklump being a plant or fungi.

• Surface appearance. Described as mushroom-like. Again we immediately run into the problem of categorization being hard, because a biologist will tell you that a mushroom isn't a plant anyway. See also. This is very weak evidence in any case, because ...

• Overall appearance. Not very plant-like, as shown in Laurel's answer. Suggests that it isn't a plant.

• Mobility. Plants are usually immobile, animals are usually mobile. Indeterminate. (See the comments under Ginge's answer.) Even if we assume that the Horklump is immobile, however, there are plenty of sessile Muggle animals. We don't know whether the Horklump has any kind of larval stage or not.

• Cell structure. Indeterminate. This is a work of fantasy, after all. :-) Ditto for respiration and the presence or absence of chlorophyll.

• Food source. Plants typically make their own food from sunlight. Fungi eat dead plants, more or less. Animals usually eat plants or other animals. The Horklump eats worms, which puts it pretty firmly in the "animal" camp as far as food source goes. (Not conclusive by itself, of course, because of, e.g., Venus Flytraps or Venemous Tentacula.)

• Sensory and nervous system. Plants typically have "no or very basic ability to sense" whereas animals have a "much more highly developed nervous system". The Horklump's tentacles "search out [...] earthworms", which suggests a functional nervous system. However, the Mandrake is definitely described as a plant, and they have would appear to have disturbingly advanced nervous systems, so I think magic makes this characteristic far less useful than it is in the Muggle world for distinguishing plants and animals.

On the whole, the odds seem to be against the Horklump being a plant or fungi. It seems more likely to be either a sessile animal like a mussel or a slow-moving animal like a slug or snail, depending on how you interpret the text. But in the absence of information on the cellular structure, detailed behavioural characteristics, and life-cycle, not to mention the phylogenetic history, this really can only be a best guess.

Horklumps resembling mushrooms doesn’t mean they can or should be classified as mushrooms.

While the Horklump may resemble a mushroom, that does not necessarily mean it is a mushroom. Bowtruckles resemble small twigs, but certainly behave like creatures, not twigs.

“The Bowtruckle is a tree-guardian creature found mainly in the west of England, southern Germany, and certain Scandinavian forests. It is immensely difficult to spot, being small (maximum eight inches in height) and apparently made of bark and twigs with two small brown eyes.”
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For an example known to Muggles, stick insects are not actually sticks. From the description given, Horklumps act more like creatures than mushrooms - they have tentacles that they use to catch food, and most likely can use them to move as well.

“The Horklump comes from Scandinavia but is now widespread throughout northern Europe. It resembles a fleshy, pinkish mushroom covered in sparse, wiry black bristles. A prodigious breeder, the Horklump will cover an average garden in a matter of days. It spreads sinewy tentacles rather than roots into the ground to search for its preferred food of earthworms. The Horklump is a favourite delicacy of gnomes but otherwise has no discernible use.”
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With what is known about the Horklump, there is no particular evidence that it acts like a fungi, and their resemblance to mushrooms does not necessarily imply they should be classified as such. The Horklumps seen in the illustrated editions of the supplementary Harry Potter books show two different possibilities how something that is a creature can resemble a mushroom.

This is the depiction of Horklumps in the illustrated edition of “Fantastic Beasts and Where to Find Them”.

This is the depiction of Horklumps in the illustrated edition of “The Tales of Beedle the Bard”.

While they are too different to both be precisely accurate, and they should not necessarily be considered canonical, both of these are examples of possible ways a creature that should be classified as a creature can resemble a mushroom.