Learn Finnish: Huomasin, että hänellä on tapana nuuhkia ruokaa. - I noticed that she has a habit of sniffing food.

This is the core of the fixed habitual construction you are currently learning: [Adessive subject] + on tapana + [1st infinitive]. The word tapa means "habit" or "custom," and putting it in the essive case (-na) translates roughly to "as a habit." Literally, the phrase reads "On her is as a habit..."

Does the tense of on have to match the past tense of huomasin?

No, and the choice of tense changes the meaning! Because it says hänellä on tapana (present tense), it implies she currently still has this habit, even though you noticed it in the past. If she no longer did it, you would change the auxiliary verb to the past tense: hänellä oli tapana (she had a habit).

What is the difference between the verb nuuhkia used here and haistaa?

Haistaa means "to smell" (either catching a scent or taking a single deliberate smell). Nuuhkia is a frequentative verb meaning "to sniff" (like a dog sniffing around, or taking multiple quick sniffs). The -i- near the end of the verb is a common marker for this kind of continuous, repetitive action in Finnish.

Why is the object ruokaa in the partitive case?

It is the object of the verb nuuhkia. Because sniffing is a continuous, irresultative action (she doesn't "complete" or "consume" the food by merely sniffing it), the object rule dictates it must take the partitive case (ruokaa) instead of the accusative/genitive.