Because they were originally two different words in Old Irish, but have changed since then and sound the same today.
The word linn meaning "lake" was lind in Old Irish.
And linn in Old Irish was li(u)nn or le(i)nn. (And is leinn in Modern Scots Gàidhlig)
"Eat your dinner with us" is an order, using the Imperative, or modh ordaitheach in Irish:
Ith do dhinnéar linn
Note that there aren't any pronouns ("you" or tú or sibh) in either the English or the Irish in the imperative/modh ordaitheach. The plural version in Irish is Ithigí bhur ndinnéar linn
You don't say "Itheann tú do dinnéar linn" because do causes lenition - do dhinnéar.
You don't say "Itheann tú bhur ndinnéar linn" because tú is singular "you" but bhur is plural "your".
You say Itheann tú do dhinnéar linn when talking to just 1 person and Itheann sibh bhur ndinnéar linn when talking to two or more people.