It's not clear; three similar spells have three different wordings
Searching the PHB for "each slot level" finds 84 spells, from Aid to Witch Bolt, that increase their effects when upcast, with the increase occurring at each slot level.
On the other hand, searching on "every two slot levels" reveals all of two spells that use this phrase: spiritual weapon and flame blade.
In the case of flame blade, it is a second level spell that does 3d6 but when cast at
4th level or higher, the damage increases by 1d6 for every two slot levels above 2nd.
Here, it is clear that the two levels is a threshold effect (and not a statement of a rate of increase scaled to whole units); casting it at 3rd level will produce no change in damage.
So spiritual weapon is unique in that its effects increase every two levels, but its spell description explicitly says that the increased effects manifest at one spell level above its regular cast. Since it is unique, we don't have anything to compare it to in order to deduce its intent.
It could be that the damage simply increases by d8 every two levels, so that when cast at 3rd there is no difference to damage. If that is the intent, however, then it is simply poorly edited. If it is supposed to do no more damage at 3rd, then it should be worded like flame blade, without the possibility of confusion. Telling us it can be upcast to 3rd level is pointless, since we already know that from the general spell rules:
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. For instance, if Umara casts magic missile using one of her 2nd-level slots, that magic missile is 2nd level.
Effectively, the spell expands to fill the slot it is put into.
Some spells, such as magic missile and cure wounds, have more powerful effects when cast at a higher level, as detailed in a spell's description.
If Spiritual weapon cast at 3rd does no more damage than when cast at 2nd, then it does not, in fact, detail in its spell description what its more powerful effects are when cast at that higher level. The spell description is simply in error; mistakes in editing do happen.
By the time of XGtE, a different means of presenting a similar progression is used for Shadow Blade (2nd level)
When you cast this spell using a 3rd- or 4th-level spell slot, the damage increases to 3d8. When you cast it using a 5th- or 6th-level spell slot, the damage increases to 4d8. When you cast it using a spell slot of 7th level or higher, the damage increases to 5d8.
To the extent that XGtE comes after the PHB, this could be taken as a statement for how spiritual weapon is supposed to work, RAI, validating the view that spiritual weapon does not do extra damage at a third level cast but also supporting the idea that the text in the PHB is poorly edited and ambiguous. Note, however, that in order to be explicit about the damage done by shadow blade, XGtE had to devote five lines of text to the "At Higher Levels" description compared to the three lines for spiritual weapon in the PHB.
While I doubt that the text in the spiritual weapon spell description is correct and intentional, it can be run as written, in which case it does more damage at every spell level increase. Returning to the 84 spells that do increase in effect at "each slot level", some of the increases are not "partible"; they are measured in creatures, amount of ammunition, challenge ratings1, hp, temporary hp, spell levels, rays, or missiles. None of these can exist within the game as fractions.
However, other spell effects that increase with spell level are measured in things that are partible; gallons, feet, dice of healing, and dice of damage. Dice of damage, in fact, are by far the most common effect that increases with spell level.
If Spiritual weapon increases by 1d8 for every two levels, this is the same as saying it increases by half a d8 every level2.
Thus, when cast at 3rd level, you could have spiritual weapon do 1.5d8 plus your spell casting mod.
When cast at 4th level, the base damage is 2d8, and when cast at 5th it could be 2.5d8.
This small increase in damage is hardly unbalancing, gives the players more meaningful choices over their spell slot use, and allows you to run the spell as written if you want.
1Yes, there are challenge ratings expressed as fractions, but these are below the minimum challenge rating effects of the spells that produce increases in challenge rating with higher level casts: cf. Summon Elemental, Summon Fey.
2This is rolled as a d8, halved, and rounded down. So for example a "1" rolled on half a d8 is 0 points of damage, and a "3, 5, 4" rolled on 2.5d8 is 10 points of damage.